Interval density pressure management methods

ABSTRACT

A method for estimating equivalent top of fluid level or a theoretical surface annular back pressure in a subterranean wellbore includes acquiring first and second axially spaced pressure measurements in the wellbore. The pressure measurements may then be processed to compute the equivalent top of fluid level and/or theoretical surface annular back pressure of drilling fluid between the measurement locations. A tool string including a large number of axially spaced pressure sensors (e.g., four or more or even six or more) electronically coupled with a surface processor via wired drill pipe may be used to obtain a plurality of values corresponding to various wellbore intervals. The equivalent top of fluid level and/or theoretical surface annular back pressures may be used in automated managed pressure drilling operations.

CROSS REFERENCE TO RELATED APPLICATIONS

This application claims the benefit of U.S. Provisional Application Ser.No. 61/527,948 entitled Interpretation Methodologies and Calculationsfor Wired Drill Pipe Along String Measurements of Pressure andTemperature, filed Aug. 26, 2011.

FIELD OF THE INVENTION

Disclosed embodiments relate generally to geotechnical fieldmeasurements and more particularly to Along String Measurements (ASM)that may be incorporated in repeater hardware sections of Wired DrillPipe (WDP). Methods are disclosed for computing sequential andnon-sequential pressure and temperature measurements in these repeatersas well as pressures and temperatures measured by Bottom Hole Assembly(BHA) components. Methods are further disclosed for utilizing thesemeasurements to characterize the subterranean formations, the drillingfluid, and the drilling process.

BACKGROUND INFORMATION

During drilling operations, measurements of downhole conditions takenwhile drilling can provide valuable information that may be used to by adrilling operator to improve efficiency and performance and minimizerisk. Such measurements, when transmitted to the surface while drilling,may also provide an essentially real time view of changing downholeconditions allowing for essentially real time performance improvementsand risk avoidance. There is considerable interest in the industry inrisk avoidance since even relatively minor interruptions in drillingoperations can be prohibitively expensive.

The recent introduction of Wired Drill Pipe (WDP) has significantlyincreased the communication bandwidth between downhole measurementsensors and the surface and therefore the total quantity of data thatmay be transmitted to the surface during a drilling operation. Forexample, measurement while drilling (MWD) and logging while drilling(LWD) data, including borehole imaging data, may be readily transmittedto the surface while drilling using WDP. Along string measurements(ASM), for example, including along string pressure and temperaturemeasurements may also be transmitted to the surface during drilling.

While along string pressure and temperature measurements are known inthe art, there has been no disclosure of methods for computingsequential and non-sequential pressure and temperature intervaldensities nor any methods of utilizing such interval densities tocharacterize the subterranean formations, the drilling fluid, or thedrilling process. There remains a need in the art for furtherdevelopment.

SUMMARY

Methods for pressure management using measured interval densities aredisclosed. For example, a tool string including at least first andsecond axially spaced pressure sensors may be deployed in a subterraneanborehole. Pressure measurements may then be used to compute anequivalent top of fluid level or a theoretical surface annular backpressure between the pressure sensors (i.e., between first and secondmeasured depths in the borehole). The tool string may further include alarge number of longitudinally spaced pressure sensors (e.g., four ormore or even six or more) electronically coupled with a surfaceprocessor via wired drill pipe thereby enabling equivalent top of fluidlevel and theoretical surface annular back pressure to be measured atmultiple intervals in the wellbore.

The disclosed embodiments may provide various technical advantages. Forexample, various disclosed embodiments provide for automated backpressure control in certain managed pressured drilling operations.Changes to the applied back pressure may be made automatically inresponse to various drilling conditions, for example, including a changein cuttings density, borehole volume changes such as washout and packoff, formation fluid flowing into the wellbore, lost circulation, anddrilling fluid density changes.

In one non-limiting embodiment a method for estimating an equivalent topof fluid level in a subterranean wellbore is disclosed. The methodincludes: (a) deploying a tool string in the wellbore, the tool stringincluding first and second subsurface longitudinally spaced pressuresensors deployed at corresponding first and second measured depths inthe wellbore; (b) causing the first and second pressure sensors toacquire first and second annular drilling fluid pressure measurements atthe first and second measured depths; and (c) causing a processor toprocess the first and second pressure measurements to compute anequivalent top of fluid level for a wellbore interval between the firstand second measured depths.

In a second non-limiting embodiment a method for computing a theoreticalsurface annular back pressure in a subterranean wellbore is disclosed.The method includes: (a) deploying a tool string in the wellbore, thetool string including first and second subsurface longitudinally spacedpressure sensors deployed at corresponding first and second measureddepths in the wellbore; (b) causing the first and second pressuresensors to acquire first and second annular drilling fluid pressuremeasurements at the first and second measured depths; and (c) causing aprocessor to process the first and second pressure measurements tocompute the theoretical surface annular back pressure for a wellboreinterval between the first and second measured depths.

In a third non-limiting embodiment a method for controlling surfaceannular back pressure in managed pressure drilling operations isdisclosed. The method includes: (a) acquiring first and second annulardrilling fluid pressure measurements at first and second measured depthsin a subterranean wellbore; (b) processing the first and second pressuremeasurements to compute a theoretical surface annular back pressure fora wellbore interval between the first and second measured depths; (c)acquiring a surface annular back pressure measurement; and (d) adjustingthe surface annular back pressure such that the measured surface annularbackpressure is substantially equal to the theoretical surface annularback pressure computed in (b).

This summary is provided to introduce a selection of concepts that arefurther described below in the detailed description. This summary is notintended to identify key or essential features of the claimed subjectmatter, nor is it intended to be used as an aid in limiting the scope ofthe claimed subject matter.

BRIEF DESCRIPTION OF THE DRAWINGS

For a more complete understanding of the disclosed subject matter, andadvantages thereof, reference is now made to the following descriptionstaken in conjunction with the accompanying drawings, in which:

FIG. 1 depicts one example of a conventional drilling rig on whichdisclosed methods may be utilized.

FIG. 2 depicts a flow chart of one example of a method embodiment forobtaining an interval density of a subterranean wellbore.

FIG. 3 depicts one example of a multi-dimensional depth and time basedarray (database) including two variables.

FIG. 4 depicts modeled oil based mud (OBM) density as a function ofpressure and temperature.

FIGS. 5A and 5B depict one example of a log including computed intervaldensities obtained during an ASM while drilling operation.

FIGS. 6, 7, and 8 depict a hypothetical example of a well drillingoperation in which a change in formation lithology is encountered thatresults in a reduced cuttings density with FIG. 6 depicting thehypothetical drilling operation at time t₁=0, FIG. 7 depicting timet₂=t₁+Δt, and FIG. 8 depicting time t₃=t₂+Δt.

FIGS. 6, 9, and 10 depict a hypothetical example of well drillingoperation in which a portion of the borehole becomes enlarged during thedrilling operation with FIG. 6 depicting the hypothetical drillingoperation at time t₁=0, FIG. 9 depicting time t₂=t₁+Δt, and FIG. 10depicting time t₃=t₂+Δt.

FIGS. 11, 12, and 13 depict a hypothetical example of a well drillingoperation in which borehole cuttings drop out of suspension and form apack-off with FIG. 11 depicting the hypothetical drilling operation attime t₁=0 and FIGS. 12 and 13 depicting distinct methodologies forcomputing interval densities at time t₂=t₁+Δt.

FIGS. 14, 15, 16, and 17 depict a hypothetical example of a welldrilling operation including a formation fluid inflow event (alsoreferred to as a kick) with FIG. 14 depicting the hypothetical drillingoperation at time t₁=0, FIG. 15 depicting time t₂=t₁+Δt, FIG. 16depicting time t₃=t₂+Δt, and FIG. 17 depicting time t₄=t₃+Δt.

FIG. 18 depicts one example of a visual display illustrating inflow as afunction of time and depth.

FIGS. 14, 19, and 20 depict a hypothetical example of a well drillingoperation including a drilling fluid outflow event with FIG. 14depicting the hypothetical drilling operation at time t₁=0 and FIGS. 19and 20 depicting time t₂=t₁+Δt. FIG. 20 differs from FIG. 19 in that thedrilling fluid level has dropped below the first ASM.

FIGS. 21, 21A and 21B depict one example of a log from a well drillingoperation in which drilling fluid flowed out of the wellbore into theformation.

FIGS. 22A and 22B depict schematic depth versus pressure plots thatillustrate equivalent top of fluid level changes that may result fromlost circulation events.

FIGS. 23, 23A and 23B depict another example of a log from the welldrilling operation depicted on FIGS. 21, 21A and 21B.

FIGS. 24, 24A and 24B depict still another example of a log from thewell drilling operation depicted on FIGS. 21, 21A and 21B.

FIGS. 25 and 26 depict a hypothetical example of a well drillingoperation in which cuttings are dropping out of suspension in theannular drilling fluid with FIGS. 22A and 22B depicting the hypotheticaldrilling operation at time t₁=0 and FIGS. 23, 23A and 23B depicting timet₂=t₁+Δt.

DETAILED DESCRIPTION

FIG. 1 depicts a drilling rig 10 suitable for using various methodembodiments disclosed herein. A semisubmersible drilling platform 12 ispositioned over an oil or gas formation (not shown) disposed below thesea floor 16. A subsea conduit 18 extends from deck 20 of platform 12 toa wellhead installation 22. The platform may include a derrick and ahoisting apparatus for raising and lowering a drill string 30, which, asshown, extends into borehole 40 and includes a drill bit 32 deployed atthe lower end of bottom hole assembly (BHA) 50. In the depictedembodiment, drill string 30 includes a plurality of joints of wireddrill pipe and therefore provides a high bandwidth digitalcommunications channel (e.g., a bandwidth on the order of 5kilobits/sec) between the BHA 50 and the surface.

Drill string 30 includes a plurality of longitudinally spaced wireddrill pipe repeater subs 34, at least some of which include annularpressure and temperature sensors 36 and 38. These sensor containingrepeater subs may be referred to herein as XLINKS and may optionallyfurther include internal pressure and temperature sensors (not shown).It will be understood that internal sensors are configured to measurethe pressure and temperature of the drilling fluid in the drill string30 while the annular (or external) sensors are configured to measure thepressure and temperature of the drilling fluid in the annulus betweenthe drill string 30 and the borehole wall. Internal and annular pressureand temperature sensors may also be deployed within the various MWDand/or LWD tools included in the BHA 50. Example BHA pressure andtemperature sensors are depicted at 52 and 54. The aforementionedpressure and temperature sensors may be in communication with thesurface via the high bandwidth digital communications channel such thatthe along string pressure and temperature measurements may betransmitted to the surface while drilling. The pressure and temperaturesensors (or the repeater subs 34) may also include onboard memory forsaving the pressure and temperature measurements for later analysis.Other drill-string components (although not explicitly depicted) mayalso contain annular and internal pressure and temperature sensors, forexample, including EMAG repeaters, mud pulse signal boosters and,acoustic telemetry boosters. Pressure and temperature measurementsobtained via these sensors may also be transmitted to the surface whiledrilling (or stored in downhole memory) and utilized in the methodembodiments disclosed hereinbelow.

The pressure and temperature sensors may have substantially anylongitudinal spacing along the length of the drill string 30. Forexample, the spaced pressure and temperature sensors may have alongitudinal spacing in a range from about 500 to about 5000 feet inmeasured depth. Moreover, the spacing between the pressure andtemperature sensors is not necessarily uniform. For example, alongitudinal spacing between first and second sensors is not necessarilyequal to the spacing between second and third sensors. The disclosedembodiments are not limited in these regards.

The disclosed embodiments are also not limited to the use any particulartype of BHA and/or repeater sub pressure sensors. Substantially anysuitable pressure sensors may be utilized provided that they providesufficient accuracy and precision and are robust in demanding downholeenvironments. For example, pressure sensors that make use of straingauges (such as those that are commercially available from PaineElectronics, LLC) may be utilized. Likewise, silicon-on-insulator solidstate pressure gauges may also be utilized.

It will be understood that the deployment illustrated on FIG. 1 ismerely an example. BHA 50 may include substantially any suitabledownhole tool components, for example, including a steering tool such asa rotary steerable tool, a downhole telemetry system, and one or moreMWD or LWD tools including various sensors for sensing downholecharacteristics of the borehole and the surrounding formation. Thedisclosed embodiments are not limited in these regards. Moreover, thedisclosed methods may be used in wellbore applications other thandrilling application, for example, including fluid samplingapplications, well control during tripping, well maintenance, completionand production applications, and the like.

It will be further understood that disclosed embodiments are not limitedto use with a semisubmersible platform 12 as illustrated on FIG. 1. Thedisclosed embodiments are equally well suited for use with eitheronshore or offshore subterranean operations. Moreover, it will beappreciated that the terms borehole and wellbore are usedinterchangeably herein.

The foregoing detailed description is divided into two principlesections, the first describing methodologies for computing intervalgradients for along string pressure and temperature measurements. Thesecond section describes methodologies for utilizing the computedinterval gradients to interpret various formation and drilling fluidproperties and the overall drilling process.

Interval Density Computation Methodologies

FIG. 2 depicts a flow chart of one example of a method embodiment 100for determining an interval density in a subterranean wellbore. A toolstring (e.g., drill string 30 depicted on FIG. 1 or a production orcompletion string) is deployed in the wellbore at 102. The tool stringincludes at least first and second subsurface pressure sensors (e.g.,annular pressure sensors or internal pressure sensors) deployed atcorresponding first and second measured depths in the wellbore. Thepressure sensors may be used to measure corresponding first and secondpressures at 104. The first and second pressures may then be processedto obtain the interval density at 106. It will be understood that toolsstrings employing three or more pressure sensors may also be utilizedand enable a plurality of interval densities to be obtained.

The density of a fluid under static conditions within the intervalbetween two pressure measurements may be computed from knowledge of avertical spacing between the pressure sensors and the actual pressuremeasurements. A temperature gradient can likewise be computed. Ingeneral, given a number n spaced apart pressure measurements, acorresponding number of intervals between all sensor combinations(neighbor and otherwise) may be computed, for example, as follows:Number of Intervals=Σ_(i=1) ^(i=n−1)(n−i)  Equation 1

For example, given 2 spaced apart sensors, 1 interval is available;given 3 spaced apart sensors, 3 intervals are available; given 4 spacedapart sensors, 6 intervals are available, given 5 spaced apart sensors,10 intervals are available, and so on. In certain of the disclosedmethod embodiments the number of interval densities computed N may, forexample, be in the range: n−1≦N≦Σ_(i=1) ^(i=n−1)(n−i).

Utilizing any one annular pressure measurement, a density of a fluid(e.g., drilling fluid) under static conditions in a wellbore may becomputed, for example, as follows:

$\begin{matrix}{{{Annular}\mspace{14mu}{density}_{avg}} = {{\left( \frac{P}{Z_{md}{\cos({Inc})}} \right)C_{1}} = {\left( \frac{P}{TVD} \right)C_{1}}}} & {{Equation}\mspace{14mu} 2}\end{matrix}$

where the annular density represents an average density of the annularfluid (e.g., in pounds per gallon), P represents the annular pressure(e.g., in psia), Z_(md) represents the measured depth of the well, TVDrepresents the true vertical depth of the well, Inc represents theaverage borehole inclination, and C₁ represents a units conversionconstant (e.g., 19.25 ppg/psi/ft).

It will be understood by those of ordinary skill in the art that thedensity of a fluid may be expressed in various units. The commonoilfield unit of pounds per gallon is given in Equation 2. Equivalentvertical head may be used to express the pressure in terms of thevertical height of a column of fluid and may be computed as follows:

$\begin{matrix}{{{Equivalent}\mspace{14mu}{Vertical}\mspace{14mu}{Head}} = \frac{{PC}_{1}}{{density}_{avg}}} & {{Equation}\mspace{14mu} 3}\end{matrix}$

where, as is known to those of ordinary skill in the art, vertical headrefers to hydraulic head (e.g., in units of feet).

Measured Annulus Interval Circulating Density

Of particular interest in this disclosure are methods for computinginterval densities (i.e., the density of the fluid) between variousspaced apart sensors (e.g., between first and second sensors or betweenfirst, second, and third sensors). Utilizing the pressure measurementsassociated with the endpoints of a specific interval, the density of afluid between the two sensors may be computed for various specific casesaccording to the following methodologies. For example, the intervaldensity of a circulating fluid may be computed as follows:

$\begin{matrix}{{MA\_ ICD}_{avg} = {{\frac{\Delta\; P}{\Delta\;{TVD}}C_{1}{MA\_ ICD}_{avg}} = {\quad{\left\lbrack \frac{\left( {P_{n + 1} - P_{n}} \right)C_{n}}{\left( {Z_{{MD}{({n + 1})}} - Z_{{MD}{(n)}}} \right){\cos({Inc})}} \right\rbrack = \left\lbrack \frac{\left( {P_{n + 1} - P_{n}} \right)*C_{1}}{\left( {Z_{{TVD}{({n + 1})}} - Z_{{TVD}{(n)}}} \right)} \right\rbrack}}}} & {{Equation}\mspace{14mu} 4}\end{matrix}$

where MA_ICD represents an averaged measured annulus intervalcirculating density, ΔP represents a change in pressure between firstand second measured depths, ΔTVD represents a change in true verticaldepth between the first and second measured depths, P_(n) and P_(n+1)represent annular pressure measurements at the first and second depths nand n+1, Z_(MD(n)) and Z_(MD(n+1)) represent the first and secondmeasured depths, and Z_(TVD(n)) and Z_(TVD(n+1)) represent the truevertical depths of the first and second measured depths. Those ofordinary skill in the art will readily appreciate that the true verticaldepth (or a change in true vertical depth) may be represented by themeasured depth (or a change in measured depth) times the cosine of theaverage wellbore inclination within an interval.

Under dynamic conditions, e.g., when circulating drilling fluid during adrilling operation, MA_ICD includes the effects of temperature on thecompressibility of the input drilling fluid, absolute pressure effectson the density, the volume and mass of the suspended cuttings, theinflow or outflow of drilling fluid between the sensors, and thefrictional pressure losses of the circulating mud. This computedinterval density (MA_ICD) is described in more detail below via variousplots and comparisons with other computed interval densities (e.g., inFIGS. 6 through 26).

Measured Annulus Interval Static Density

Interval densities may also be computed during non-circulating (static)conditions as well using Equation 4. Such conditions are generallyavailable at every connection while adding a pipe stand or a joint tothe drill string and occasionally while drilling is suspended during thedrilling of a stand. Under such static conditions, the annularfrictional pressure losses are absent and the only effects on theinterval densities are pressure, temperature, and suspended cuttingseffects. This parameter is referred to as MA_ISD and is computed usingEquation 4 but under static, non-circulating conditions.

A interval static density may also be computed by subtracting modeled ormeasured frictional pressure losses from MA_ICD as computed in Equation4 when computed under circulating conditions. This approach enables asubstantially continuous determination of the interval static densityand is referred to as MA_ISD_(mf). Equation 4 may be modified to includethese frictional pressure terms as shown below in Equation 5.

$\begin{matrix}{{MA\_ ISD}_{mf} = \left\lbrack \frac{\left( {\left( {P_{n + 1} - P_{f_{n + 1}}} \right) - \left( {P_{n} - P_{f_{n}}} \right)} \right)*C_{1}}{\left( {Z_{{TVD}{({n + 1})}} - Z_{{TVD}{(n)}}} \right)\;} \right\rbrack} & {{Equation}\mspace{14mu} 5}\end{matrix}$

where P_(f) _(n) represents the frictional pressure loss acting on thefluid above the sensor n and P_(f) _(n+1) represents the frictionalpressure loss acting on the fluid above the sensor n+1.

Two methods for computing the frictional pressure loss are disclosed; ahydraulically modeled method and an in-situ measurement method. Thehydraulic model makes use of various known or estimated fluid and boreproperties to compute the frictional pressure loss. The properties mayinclude, for example, temperature, pressure, compressibility, viscosity,flow rate, and flow regime of the drilling fluid, the annular volume ofthe borehole, the borehole diameter and shape, rotation rate effects,and properties of the borehole wall such as smoothness.

The measurement method may compute the interval density, for example,using Equation 4 under non-pumping static conditions for distinct holesections or intervals in the well as a function of time. After the pumpsare turned back on and before drilling resumes this quantity may be usedin the left hand side of Equation 5 along with the measured pressures tocompute P_(f) _(n−1) −P_(f) _(n) for each distinct hole section in thewell. The dynamic frictional pressure loss is generally a strongfunction of the flow rate and rotation rate for a given hole section andperiod of time during the drilling of the well. Therefore, this pressureloss is generally a slowly varying value with time under steady stateflow conditions. For example, it may be in the range from 0.1 to 1 poundper gallon in a 10,000 foot vertical well. In this second method, anin-situ determination of frictional pressure loss only needs to beperformed periodically as long as the drilling parameters do not change(e.g., rotation rate, flow rate, and the BHA components in each distincthole section that may have a different frictional pressure loss). Whenthe drilling parameters change, the second method may be repeated.

In practice it may be advantageous to make use of both the theoreticaland measurement methodologies for computing the frictional pressurelosses. For example, when the two methods give similar values, thehydraulic model may be used with increased confidence. Differencesbetween the measured and modeled frictional losses may also be used tocalibrate the hydraulic model, compute a cuttings density, or flagcertain drilling events of interest as described in more detail below.

Upon determining the frictional pressure losses, the measured annulusinterval static density MA_ISD_(mf) may be determined while circulatingand drilling by substituting the frictional pressure losses intoEquation 5. The MA_ISD_(mf) may be computed at various time intervalsduring drilling.

It should be understood that in drilling operations in which backpressure is applied to the annular fluid (e.g., as is done duringmanaged pressure drilling (MPD) applications), Equations 4 and 5 do notrequire a back pressure term since a differential pressure is used todetermine the interval density. It should also be understood that theinterval gradients are a direct function of a down-hole pressure anddepth measurements. Therefore any of the principles applied to theinterval gradient computations apply to pressure measurements, whethermeasured or theoretical.

Density of Constituent Mud Components

The measured annulus interval static density MA_ISD or computedMA_ISD_(mf) may be taken to be the sum of the individual densities ofthe individual components of the static annular fluid which may be validfor non-soluble components such as liquid formation fluids and formationcuttings normally encountered during drilling. This may be expressedmathematically, for example, as follows and may enable individualcomponent specific gravities to be computed when their volumetricpercentages are known:

$\begin{matrix}{{MA\_ ISD}_{{avg}\;} = {\sum\limits_{i = 1}^{i = n}\left( \frac{M_{i}}{V_{i}} \right)}} & {{Equation}\mspace{14mu} 6}\end{matrix}$

where MA_ISD_(avg) represents an average measured annular intervalstatic density, M_(i) represents the mass of non-soluble component i,and V_(i) represents the volume of non-soluble component i. MA_ISD_(avg)may also be expressed as a volume weighted average of the individualconstituents in the drilling fluid mud. It should be noted that theproduct of volume and density also represents the mass and may thereforebe re-written in terms of volumetric percentages as follows:

$\begin{matrix}{{MA\_ ISD}_{mixture} = {\sum\limits_{i = 1}^{i = n}\left( \frac{V_{i}*{SG}_{i}}{V_{mixture}} \right)}} & {{Equation}\mspace{14mu} 7}\end{matrix}$

where MA_ISD_(mixture) represents the measured annular interval staticdensity of a mixture, V_(i) represents the volume of non-solublecomponent i, V_(mixture) represents the total volume of the mixture, andSG_(i) represents the density (or specific gravity) of component i.

The drilling fluid flowing towards the surface in the annulus generallyincludes a combination of the drilling fluid that is pumped downwardthrough the interior of the drill pipe and cuttings removed by the drillbit during drilling. The volumetric flow rate in the annulus may beexpressed as a combination of these two expected constituents plus anadditional term that quantifies increased or reduced flow owing to theaddition of an unexpected or unwanted constituent or the loss of aconstituent. The additional term may quantify, for example, an inflow offormation fluid into the annulus or an outflow of drilling fluid intothe formation. The inflow or outflow may involve either previouslydrilled or currently drilled formations. Alternatively, the additionalterm may quantify additional cuttings spalling off the borehole wallafter drilling. ASM and corresponding interval density computations mayenable the enable these inflow or outflow constituents to be identifiedand located along the length of the borehole.

As stated above, the annular drilling fluid includes a combination ofthe drilling fluid that is pumped downward through the interior of thedrill pipe and cuttings removed by the drill bit. The cuttings volumemay be accounted for by integrating the flow rate in a unit volume ofannular fluid over a specified time interval and recognizing that theflow rate out of the unit volume must equal the flow rate into the unitvolume. In other words, the flow rate of the mixture may be set equal tothe sum of the individual flow rates into this volume. The accumulatedvolume of the mixture flowing out of the unit annular volume over agiven time period may be expressed mathematically, for example, asfollows:

$\begin{matrix}{{\int_{t\; 1}^{t\; 2}{Q_{mixture}{\mathbb{d}t}}} = {{\int_{t\; 1}^{t\; 2}{Q_{out}{\mathbb{d}t}}} = {\int_{t\; 1}^{t\; 2}{\left( {Q_{{mud}\mspace{14mu} i\; n} + Q_{cuttings} + Q_{x}} \right){\mathbb{d}t}}}}} & {{Equation}\mspace{14mu} 8}\end{matrix}$

where Q_(mixture) represents the volumetric flow rate of the mixture attime t, Q_(out) represents the volumetric flow rate out of the unitannular volume, Q_(mud in) represents the volumetric flow rate ofdrilling fluid (mud) pumped into the unit annular volume at time t,Q_(cuttings) represents the volumetric flow rate of cuttings flowinginto the unit annular volume at time t, and Q_(x) represents thevolumetric flow rate of component x flowing in or out of the unitannular volume at time t. Q_(mud in) and Q_(cuttings) may be furtherdefined, for example, as follows:

$\begin{matrix}{Q_{{mud}\mspace{14mu} i\; n} = {{{TFLO}\left( \frac{gal}{\min} \right)}*\left( \frac{60\mspace{14mu}\min}{1\mspace{14mu}{hr}} \right)*\left( {0.1337\frac{{ft}^{3}}{gal}} \right)}} & {{Equation}\mspace{14mu} 9}\end{matrix}$

where TFLO represents the drilling fluid flow rate in units of gallonsper minute. TFLO may be determined at the surface using methods known tothose of ordinary skill in the art, for example, using the rig pumpstroke rate, number of pump cylinders in use, their displacement/stroke,and the pump efficiency. When pumping a compressible fluid such assynthetic oil-based mud (SOBM), the down-hole flow rates tend to changedue to pressure and temperature effects on the fluid properties. Themeasured ASM pressures and temperatures of the interior drill pipe fluidproperties may be used to measure the fluid temperature and density inthe drill pipe in order to determine the in-situ fluid compressibilityand from this calculate the actual down-hole flow rate given the surfaceflow rate. The downhole flow rate may also be measured downhole.

The volume rate of cuttings being created and flowing into the annulusduring the drilling operation may be considered an input variable andmay be expressed mathematically, for example, as follows:

$\begin{matrix}{Q_{cuttings} = {\pi*r^{2}*{{ROP}\left( \frac{ft}{hr} \right)}\left( {1 - {K*\phi}} \right)}} & {{Equation}\mspace{14mu} 10}\end{matrix}$

where r represents the borehole radius, ROP represents the drilling rateof penetration, K represents percentage of formation porosity destroyedby the crushing action of the bit, and φ represents the formationeffective porosity.

The percentage of formation porosity destroyed by the action of the bitK may be estimated by observing the size of the cuttings while drilling.When K is set to unity, the crushing action of the bit destroys all ofthe porosity, creating cuttings akin to individual sand grains. Forexample, in unconsolidated sands, the cuttings size will be small andfew present with predominantly individual sand grains seen in thesamples caught coming from the shale shakers. In shale formations,competent or cemented rock, K is typically less than unity due to thecrushing component of the bit being reduced (or minimized depending uponthe hardness of the formation).

Determining a value of K may be advantageous in certain drillingoperations, for example, when a driller desires to compute an expectedvolumetric flow rate of cuttings in certain cuttings management programsthat determine the volume of cuttings that remain in the borehole andmay potentially restrict the movement of the BHA. However, in certainapplications it may be sufficient to set K to unity so as to haveQ_(cuttings) represent the matrix or rock volume of the formation. Thisallows the density of the fluid contained within the pore volume to beaccounted separately in Equation 11.2 as described in more detail below.

The formation porosity φ may be estimated, for example, from anormalized rate of penetration (ROP) as disclosed in U.S. Pat. No.4,949,575 or in Rasmus and Stephens (SPE Paper 20443, Real-TimePore-Pressure Evaluation From MWD/LWD Measurements and Drilling-DerivedFormation Strength). However, a fractional volume of fine grainedclay/shale/silt in the formation, V_(shale), is generally required forthis determination. V_(shale) is normally computed from LWD measurementssuch as natural gamma ray measurements, however, such LWD measurementsare not generally available at the bit.

In certain applications, a dimensionless torque (T_(D)), obtained, forexample, from a Mechanical Efficiency Log may be used to differentiatebetween drilling a porous formation and a shale formation due to theunique and increased dimensionless torque signature of a porousformation as compared to shale. Such differentiation can commonly bemade regardless of drill bit type. One example of a MechanicalEfficiency Log is given in Equation 11. V_(shale) may be estimated fromT_(D) and a dimensionless rate of penetration (R_(D)) by realizing thatboth T_(D) and R_(D) are functions of clay volumes and effectiveporosity regardless of the wear conditions of the bit (see Burgess,Falconer, and Sheppard, “Separating Bit and Lithology Effects FromDrilling Mechanics Data”, SPE 17191, 1988). Such V_(shale) measurementsmay then be updated once LWD data above the bit measures the formationproperties. T_(D) and R_(D) may be expressed mathematically, forexample, as follows:

$\begin{matrix}{T_{D} = \frac{12*{DTOR}}{{DWOB}*{BS}}} & {{Equation}\mspace{14mu} 11}\end{matrix}$

Where DTOR represents a downhole or surface measured torque, DWOBrepresents a downhole or surface measured weight on bit, and BSrepresents a drill bit diameter.

$\begin{matrix}{R_{D} = \frac{{ROP}*0.2}{{RPM}*{BS}}} & {{Equation}\mspace{14mu} 11.1}\end{matrix}$

Where ROP represents a rate of penetration and RPM represents a rotationrate of the drill string in revolutions per minute.

The pore fluid contained within the pore space of the formation may beretained within the cutting chip or released into the annular fluiddepending on the crushing factor, K. Regardless of the degree ofcrushing, it will affect the measured interval densities of the annularfluid and may therefore be accounted for separately.

$\begin{matrix}{Q_{{pore}\;\_\;{fluid}} = {\pi*r^{2}*{{ROP}\left( \frac{ft}{hr} \right)}(\phi)}} & {{Equation}\mspace{14mu} 11.2}\end{matrix}$

where Q_(pore) _(_) _(fluid) represents the pore fluid volumetric flowrate into the annulus in units of cubic feet per hour, r represents theborehole radius, ROP represents the rate of penetration, and φrepresents the formation effective porosity.

The drilling fluid (mud) flow rate exiting the annulus at the surface,Q_(mixture) or Q_(out), may also be considered an input measurablevolume and may be measured, for example, by a paddle-type measurementplaced into the flow out line or by a venturi-type measurement or othermeans when utilizing managed pressure drilling (MPD) type equipment.This leaves the quantity Q_(x) as the only unknown in Equation 8. Indrilling operations this represents one way of detecting a formationfluid inflow or a “kick” (as it is referred) in the industry. However,under conditions in which Q_(x) has been verified to be approximatelyequal to zero (e.g., via stopping the mud pumps and performing a flowcheck), Equation 8 may alternatively be used to measure the volume ofcuttings flowing into the annulus.

However, in certain applications it can be difficult to utilize theabove described methodology to determine Q_(x) given measurements ofQ_(cuttings), Q_(mud) _(_) _(in), and Q_(mixture) or Q_(out). This canbe due to large variations in mud flow volumes sometimes seen whiledrilling which can in turn be due to erratic pump strokes, fluidcompressibility, and inaccurate sensor measurements of these quantities.Equation 10 is often the most accurate means of determining the cuttingsvolumes. Knowing the volume of cuttings generated and keeping track ofthe volume of cuttings exiting the wellbore allows one to determine thevolume of cuttings, if any, that have been left in the borehole.

However, it is desirable to not only know the volume of cuttings beinggenerated, but the density of the cuttings in the annulus between anytwo ASM pressure measurements since this gives us information as to thetype of formation being drilled. Within any two or more arbitrary depthsin the annulus, the relative volumetric percentage of the cuttingsvolume in the annulus makes up a larger percentage than that computed byEquation 8 due to the cuttings travelling upward through the annulus ata lower velocity than that of the drilling fluid. A corrected cuttingsvolume may be computed by considering a “slip” velocity for the cuttingswhere V_(slip)=V_(annular)−V_(cuttings). A transport efficiency F_(T)_(_) _(cuttings) may be defined as the ratio of the cuttings velocity tothe average mud annular velocity and may be expressed mathematically,for example, as follows:

$\begin{matrix}{F_{T\;\_\;{cuttings}} = {\frac{{VEL}_{cuttings}}{{VEL}_{mixture}} = \frac{\left( \frac{Q_{cuttings}}{{Area}_{annulus}*f_{cuttings}} \right)*\left( {{\cos({Incl})} + {a*{\sin({Incl})}}} \right)}{\left( \frac{Q_{{mud}\mspace{14mu} i\; n} + Q_{x} + Q_{{pore}\;\_\;{fluid}}}{{Area}_{annulus}*\left( {1 - f_{cuttings}} \right)} \right)}}} & {{Equation}\mspace{14mu} 12}\end{matrix}$

where f_(cuttings) represents the volumetric fraction of cuttings in themud flowing in the annulus, Area_(annulus) represents the crosssectional area of the annulus a particular depth Z, Q_(mud) representsthe volume flow rate of mud from Equation 9, Q_(cuttings) represents thevolume flow rate of cuttings from Equation 10, Q_(pore) _(_) _(fluid)represents the volume flow rate of pore fluid from Equation 11.2, and arepresents a saltation flow transport partitioning coefficient, which isgenerally a function of RPM and Q_(mixture).

The transport efficiency can be computed from empirical correlationssuch as those disclosed in (i) Sifferman, et al., “Drill CuttingTransport in Full-Scale Vertical Annuli,” J. Pet. Tech., November 1974,1295-1302; (ii) Moore, “Drilling Practices Manual,” Petroleum PublishingCo., Tulsa, 1974, and (iii) Sample and Bourgoyne, “Development ofImproved Laboratory and Field Procedures for Determining the CarryingCapacity of Drilling Fluids,” SPE 7497, 1978. The volumetric fraction ofcuttings flowing in the annulus is also a function of wellboreinclination since the cuttings tend to fall out of suspension in highinclination sections. The constant a is used to account for the factthat as the wellbore becomes closer to horizontal, the cuttings tend todrop out of suspension and are transported along the wellbore in a“saltation” type mechanism. The inclination and saltation terms inEquation 12 are intended to result in a net upward or vertical cuttingsslip velocity. Equation 12 may then be rearranged to compute the termf_(cuttings), for example, as given in Equation 13.

$\begin{matrix}{f_{cuttings} = \frac{X*Q_{cuttings}}{\begin{matrix}{{X*Q_{cuttings}} + {F_{T\;\_\;{cuttings}}*}} \\\left( {Q_{{mud}\mspace{14mu} i\; n} + Q_{x} + Q_{{pore}\;\_\;{fluid}}} \right)\end{matrix}}} & {{Equation}\mspace{14mu} 13}\end{matrix}$

where X=cos Inc+a sin Inc.

Being liquid at downhole temperatures and pressures, the formation porefluid volume that is released into the annulus may have negligible slipvelocity with respect to the mud. The fractional volume of the porefluid f_(pore) _(_) _(fluid), mud f_(mud) _(_) _(in), and influx/outfluxmaterial f_(x) may then be given, for example, as follows in Equation13.1, 13.2, and 13.3.

$\begin{matrix}{f_{pore\_ fluid} = \frac{X*F_{T\_ cuttings}*Q_{pore\_ fluid}}{\begin{matrix}{{X*Q_{cuttings}} + {F_{T\_ cuttings}*}} \\\left( {Q_{{mud}\mspace{14mu} i\; n} + Q_{x} + Q_{pore\_ fluid}} \right)\end{matrix}}} & {{Equation}\mspace{14mu} 13.1} \\{f_{mud\_ in} = \frac{X*F_{T\_ cuttings}*Q_{mud\_ in}}{\begin{matrix}{{X*Q_{cuttings}} + {F_{T\_ cuttings}*}} \\\left( {Q_{{mud}\mspace{14mu} i\; n} + Q_{x} + Q_{pore\_ fluid}} \right)\end{matrix}}} & {{Equation}\mspace{14mu} 13.2} \\{f_{x} = \frac{X*F_{T\_ cuttings}*Q_{x}}{\begin{matrix}{{X*Q_{cuttings}} + {F_{T\_ cuttings}*}} \\\left( {Q_{{mud}\mspace{14mu} i\; n} + Q_{x} + Q_{pore\_ fluid}} \right)\end{matrix}}} & {{Equation}\mspace{14mu} 13.3}\end{matrix}$

In some applications, especially at shallower depths, the formation porefluid volume f_(pore) _(_) _(fluid) that is released into the annulusmay have a slip velocity with respect to the mud velocity when there aredensity differences between the two fluids. This slip velocity cangenerally be computed and made available from a hydraulics module incommercial borehole cleaning or cuttings management programs.

Transformation from volumetric or fractional flow dimensions to a depthdimension requires the simultaneous consideration of cross-sectionalareas and fractional volumes. The annular volume may be representedmathematically, for example, as follows:

$\begin{matrix}{{Vol}_{annulus} = {\int_{z = n}^{z = {n + 1}}{\pi*r^{2}*\left( {D_{bh}^{2} - D_{p}^{2}} \right){\mathbb{d}z}}}} & {{Equation}\mspace{14mu} 14}\end{matrix}$

where V_(annulus) represents the annular volume between any two depthsz=n and z=n+1, D_(bh) represents the borehole diameter obtained forexample from the bit diameter or LWD caliper measurements, and D_(p)represents the diameter of the drill pipe located between z=n and z=n+1.Equation 14 assumes a borehole having a circular cross section. Thisassumption may be suitable for many drilling operations, however, thedisclosed embodiments are not limited in this regard. For example, amore general elliptical shape may be utilized.

It will be understood that Equation 14 is expressed in terms of boreholedepth rather than time. It will further be understood that the linkbetween the volumes and depth is the annular velocity of the mud andcuttings mixture, while the link between the depth based annular volumeand time is the rate penetration. Thus the annular volumes and fluidflow rates may be expressed alternatively as functions of time or depth.For example, the cuttings and fluid flow velocity may be integrated overa specific time period to determine the cuttings as a function of depth.

In one workflow example, an array of annular volume over discrete depthintervals may be computed using Equation 14. The array may be as fine asa few inches in depth or as sparse as one to two feet in depth. In thelower BHA (below an LWD caliper tool), the bit size may be used as theborehole diameter. The diameter may be updated using measured valueswhen LWD caliper measurements become available at the predefined depths.The diameter of the drill pipe may also be continually updated usingdiscrete functions of time as the various pipe diameters pass throughthese same depth points and the various cuttings are lifted from the bitface and carried into the annular volume. The terms Q_(mud in) andQ_(cuttings) may be computed from Equations 9 and 10 at discrete timeintervals (e.g., every few seconds). These volumes may then be utilizedin Equation 13 to compute the fractional volume of cuttings within eachdiscrete time period. The velocity of the cuttings may be integrated toobtain the corresponding depth position of the cuttings with time andmay be expressed mathematically, for example, as follows:

$\begin{matrix}{{\Delta\; Z} = {{\int_{T\; 1}^{T\; 2}{{VEL}_{cuttings}{\mathbb{d}t}}} = {\int_{T\; 1}^{T\; 2}\left( \frac{Q_{cuttings}}{{Area}_{annulus}*f_{cuttings}} \right)}}} & {{Equation}\mspace{14mu} 15}\end{matrix}$

It may improve accuracy to integrate the mud annular velocity (asopposed to or in addition to the cuttings velocity) due to the higherfractional volume and larger volumetric flow rates. This may beexpressed mathematically, for example, as follows:

$\begin{matrix}{{\Delta\; Z} = {{\int_{T\; 1}^{T\; 2}{{VEL}_{mixture}{\mathbb{d}t}}} = {\int_{T\; 1}^{T\; 2}\left( \frac{Q_{{mud}\mspace{14mu} i\; n} + Q_{x} + Q_{pore\_ fluid}}{{Area}_{annulus}*\left( {1 - f_{cuttings}} \right)} \right)}}} & {{Equation}\mspace{14mu} 16}\end{matrix}$

Equations 15 and/or 16 may be used to generate multi-dimensional arraysindexed by depth increments. Each column represents one chosen timeinterval and may contain TIME, as well as Area_(annulus), Q_(mud in),Q_(cuttings), Q_(pore) _(_) _(fluid), VEL_(cuttings), VEL_(mixture),f_(pore) _(_) _(fluid) and f_(cuttings). The total time required tocirculate the cuttings out of the annulus to the surface dictates thetotal number of time intervals (steps). For example, if a time intervalof 5 seconds is utilized and it takes 1 hour to circulate cuttings fromthe bit to the surface, then the array includes 720 times intervals(3600 sec/5 sec). Additional time intervals may be included toaccommodate periods of non-circulation (e.g., a time period in which anew pipe stand is added to the drill string). One example of amulti-dimensional depth and time based array (database) includingmultiple variables is depicted on FIG. 3. For ease of illustration onlytwo of the many variables are shown in the depicted example. It will beunderstood that rows are typically added to the array as the wellbore isdrilled deeper into the earth.

The quantities MA_ISD and MA_ICD described above and calculated usingthe ASM data and Equation 5 may include multiple depth intervals withinthe previously described depth array. This multi-dimensional array maybe integrated over the depth intervals corresponding to the ASM intervalto derive an averaged density of the mixture which may be compareddirectly with the ASM measured values. A similar process may also befollowed for the fractional cuttings volume. From Equation 7,MA_ISD_(mixture) may be expressed mathematically, for example, asfollows:MA_ISD_(mixture) =f _(cuttings)·SG_(cuttings) +f _(pore) _(_)_(fluid)·SG_(pore) _(_) _(fluid) +f _(mud in)·SG_(mud in) +f_(x)·SG_(x)  Equation 17

where f_(cuttings), f_(pore) _(_) _(fluid), f_(mud in), and f_(x)represent fractional volumes of the cuttings, pore fluid, the drillingmud, and the inflow or outflow constituents and SG_(cuttings), SG_(pore)_(_) _(fluid), SG_(mud in), and SG_(x) represent the specific gravitiesof the cuttings, pore fluid, the drilling mud, and the inflow or outflowconstituents. Under conditions in which there is no inflow, outflow orother event, such that constituent x is zero, Equation 17 may be used tocompute SG_(cuttings) as all other variables may be determined via othermeans as described above. Such calculations are described in more detailbelow.

Equation 17 may be further expanded by considering the pore fluid toinclude a combination of hydrocarbons and water that may or may not havebeen flushed by the drilling mud. The expanded form of Equation 17 maybe represented mathematically, for example, as follows:MA_ISD_(mixture) =f _(cuttings)·SG_(cuttings) +F·f _(pore) _(_) _(fluid)+S _(w)·SG_(pore) _(_) _(free) _(_) _(water) +F·f _(pore) _(_)_(fluid)·(1−S _(w))·SG_(pore) _(_) _(hydrocarbon)+(1−F)f _(pore) _(_)_(fluid)SG_(mud) _(_) _(in) +f _(mud in)·SG_(mud in) , f_(x)·SG_(x)  Equation 17.1

where F represents a flushing factor such that 1≦F≦0 with F=1representing no flushing and F=0 representing complete flushing, S_(w)represents pore water saturation, 1−S_(w) represents pore hydrocarbonsaturation, SG_(pore) _(_) _(free) _(_) _(water) represents the densityof the pore water, SG_(pore) _(_) _(hydrocarbons) represents the densityof the pore hydrocarbons, and SG_(mud) _(_) _(in) represents the densityof the input drilling fluid (mud).

When drilling under conditions of no influx or outflux (i.e., f_(x)=0),Equation 17.1 includes four unknowns (SG_(cuttings), F, S_(w), andSG_(pore) _(_) _(hydrocarbons)) with the remainder of the variablesbeing measured directly or computed from other measurements. Asdescribed above, a MEL may be used to determine whether the drilledformation is shale or a porous formation. When drilling shale, the watersaturation may be assumed to be 100%. In certain geological environmentsthe lithology of a porous formation is known to be, for example,sandstone, limestone, or dolomite such that the SG_(cuttings) can beinput. Equation 17.1 may be rearranged to solve for S_(w) as follows(recognizing that S_(hyr)=(1−S_(w)):

                                Equation  17.2$S_{w} = \frac{\begin{matrix}{{MA\_ ISD}_{mixture} - {f_{cuttings} \cdot {SG}_{cuttings}} - {f_{{mud}\mspace{14mu} i\; n} \cdot {SG}_{{mud}\mspace{14mu} i\; n}} -} \\{{\left( {1 - F} \right) \cdot f_{pore\_ fluid} \cdot {SG}_{{mud}\mspace{14mu} i\; n}} -} \\{F \cdot f_{pore\_ fluid} \cdot {SG}_{pore\_ hydrocarbons}}\end{matrix}}{F \cdot {f_{pore\_ fluid}\left( {{SG}_{{pore\_ free}{\_ water}} - {SG}_{pore\_ hydrocarbons}} \right)}}$

Given that Equations 17.1 and 17.2 include at least four unknowns,various techniques may be utilized to determine which water saturationis appropriate. For example, by assuming no flushing (F=1), inputtingSG_(cuttings) from the known lithology (e.g., shale or porous formationas determined by MEL), and assuming a value for SG_(pore hydrocarbons),enables S_(w) to be computed for various scenarios. An appropriatescenario may be selected based on expected values of S_(w). In onescenario, it may be assumed that hydrocarbons are present but that theformation is water bearing. In such a scenario the calculated watersaturation would be expected to be unity. In another scenario, it may beassumed that hydrocarbons are present and that the formation ishydrocarbon bearing. In such as scenario, the calculated watersaturation would be expected to range between 0 and 1, but typicallygreater than 0.1-0.2.

Computing S_(w) requires that the hydrocarbon density be input. Sincethis quantity is unknown, S_(w) may be computed based on a firsthydrocarbon density representing gas (SG_(gas)≈0.2) and a secondhydrocarbon density representing oil (SG_(oil)≈0.8). When the formationis gas bearing, the computed S_(w) using SG_(oil) is typically less thanzero and therefore erroneous. When the formation is oil bearing, thecomputed S_(w) using SG_(gas) is typically between zero and one, buterroneously high. The computed S_(w) using SG_(gas) advantageouslyrepresents an upper bound on the actual water saturation.

When inflow is detected, the quantity f_(cuttings)SG_(cuttings) may beassumed to be constant for a time interval. Equation 17 may then be usedto compute f_(x)SG_(x) from which SG_(x) may be computed when f_(x) isknown (e.g., from Equation 8). Determining (or estimating) SG_(x) can beadvantageous in determining the type of fluid inflow into the wellbore.

Measured Drill Pipe Internal Interval Static and Circulating Density

The aforementioned internal ASM pressure sensors that are deployed andconfigured to measure an internal pressure of the drill pipe(ASM_(internal P)) may be used to obtain internal fluid gradients withinthe drill pipe under no flow (MIF_ISD) and flowing conditions (MIF_ICD),for example, using Equation 4. The difference between MIF_ISD andMIF_ICD is generally due to frictional losses in the drill pipe. Whentwo axially spaced pressure sensors are sufficiently close to the bitand separated in TVD so as to give adequately high signal/noisemeasurements, the internal interval static density can be measured whennot pumping. The internal interval static density may also be computedusing Equations 4 and 5 as described above to determine the frictionalpressure losses and to subtract them from the measured internal dynamicinterval density. Frictional losses may also be computed using ahydraulics model.

The measured internal interval static density (MIF_ISD) is a function ofthe density of the actual fluid being pumped into the pipe at thesurface plus any pressure and temperature effects that affect thecompressibility of the fluid. If the sensor pairs are far above the bit,a computed temperature correction to the interval static density may beapplied using an appropriate hydraulics model that includes temperatureand frictional pressure effects.

MIF_ISD represents the fluid exiting the bit before any cuttings loadingand annular frictional loss effects and may therefore be used as theinput to the computation of the expected annulus fluid interval staticdensity described in more detail hereinbelow.

Expected Drill Pipe Internal Interval Static and Circulating Density

Known hydraulic modeling techniques may be utilized to predict theinternal fluid density as a function of the internal (predicted ormeasured) pressure and temperature using the surface mud densityproperties as a base fluid for the modeling. The surface mud propertiesare typically measured by mud loggers but may also be measured bysensors at the surface. Accounting for the pressure and temperatureeffects results in an expected internal fluid interval static densityEIF_ISD. By taking into account modeled frictional effects an expectedinternal fluid interval circulating density EIF_ICD may be obtained.Expected interval densities are also referred to herein as modeledinterval densities. The expected internal densities are generally equalto the measured quantities MIF_ISD and MIF_ICD when the hydraulic modelis correct. A minimization process may be used to adjust appropriatehydraulic parameters until a suitably accurate match is found.

Expected Annulus Fluid Interval Static Density

An expected annulus fluid interval static density (EAF_ISD) may beobtained by correcting MIF_ISD for pressure and temperature effects asthe input mud flows up the annulus to the surface. The EAF_ISD may becompared to the various measured interval densities to identify certainundesirable drilling events as described more detail below in variousapplications of the INTERVAL DENSITY APPLICATIONS section of thisdisclosure. The annulus pressure and temperature are typically measuredby the ASM sensors in the WDP. When these measurements are notavailable, and only the BHA sensors are present, pressure andtemperature gradients may be assumed between the BHA sensors and thesurface.

Expected Annulus Interval Static Density

The fluid leaving the bit and being pumped into the annulus is a fluidhaving properties defined by EAF_ISD, which as is described above isMIF_ISD corrected for pressure and temperature effects on the density.The cuttings load (with Q_(x)=0) computed using one or more of Equations8-16 may be added to EAF_ISD to obtain an expected annulus intervalstatic density EA_ISD. Expected interval densities are also referred toherein as ‘modeled’ interval densities. The EA_ISD represents ahypothetical fluid having the properties of the mud being injected intothe annulus at the bit loaded with the drilled and suspended cuttingshaving a certain interval density and may be expressed mathematically,for example, as follows:

$\begin{matrix}\begin{matrix}{{EA\_ ISD} = {{f_{{mud}\mspace{14mu} i\; n} \cdot {SG}_{{mud}\mspace{14mu} i\; n}} + {f_{cuttings} \cdot {SG}_{cuttings}}}} \\{= {{f_{{mud}\mspace{14mu} i\; n} \cdot {EAF\_ ISD}} + {f_{cuttings} \cdot {SG}_{cuttings}}}}\end{matrix} & {{Equation}\mspace{14mu} 18}\end{matrix}$

The difference between EAF_ISD and EA_ISD is due to the cuttingsloading. If the difference is minimal at the bottom of the hole, thecuttings density and loading effects computed using Equations 8-16 islikely correct. Given a discrepancy, the cutting density may beadjusted. If MA_ISD decreases and drops below EA_ISD as the mud flows upthe annulus into the deviated section of the borehole, it indicates thatthe cuttings may be dropping out of suspension and settling at thebottom of the borehole. Moreover, inflow or outflow from the wellboremay result in differences between these two computed parameters and maybe used to flag lost circulation and gas kicks.

Expected Annulus Interval Circulating Density

Taking the computation of EA_ISD and adding the annular frictionpressure losses to it results in an expected annulus intervalcirculating density EA_ICD. This parameter is a function of the inputmud density adjusted for temperature, pressure, cuttings load, andannular frictional pressure losses and is therefore comparable toMA_ICD. The expected and measured quantities (EA_ICD and MA_ICD) tend tobe equal to one another when the cuttings density and the frictionallosses are input correctly. When these quantities are not equal (or notclose to equal), it may indicate a change in cuttings density from theassumed cuttings density or inflow or outflow event (a Q_(x) event).EA_ICD may be expressed mathematically, for example, as follows:

$\begin{matrix}{{EA\_ ICD} = {{f_{{mud}\mspace{14mu} i\; n}{EAF\_ ISD}} + {f_{cuttings}{SG}_{cuttings}} + \frac{\left( {P_{f_{n + 1}} - P_{f_{n}}} \right) \cdot C_{1}}{\left( {Z_{{TVD}{({n + 1})}} - Z_{{TVD}{(n)}}} \right)}}} & {{Equation}\mspace{14mu} 19}\end{matrix}$

where Z_(TVD(n)) and Z_(TVD(n+1)) represent the true vertical depths ofthe well at the first and second depths n and n+1 and P_(f) representsthe frictional pressure drop acting on the fluid above the sensor asdescribed above with respect to Equations 4 and 5.

Equivalent Top of Fluid Level

The equivalent measured or true vertical depth of the top of the fluidlevel may be computed from the annular mud interval density existingbetween any two pressure sensors using the concept of hydraulic head.This may be referred to as the equivalent top of fluid level (ETOFL) andis intended to define the uppermost depth or level that a fluid wouldoccupy if it were continuous and had the same properties as the fluidbetween the two measured pressures. A back pressure may sometimes beapplied to the annular choke during managed pressure drilling (MPD)operations. With an incompressible fluid in the annulus, the pressuremay be subtracted from the measured pressure to compute ETOFL. When thefluid is compressible, simply subtracting the back pressure may not tobe suitably accurate such that it may be necessary to compute anequivalent back pressure at the sensor. Such calculations may beaccomplished, for example, using hydraulic models.

The following mathematical equations may be used to compute ETOFL in thepresence of an applied back pressure using the previously calculatedinterval densities. In these equations, a positive ETOFL indicates thatthe computed fluid level is below the surface, while a negative ETOFLindicates the fluid level is above the surface.

$\begin{matrix}{{ETOFL} = {Z_{{TVD}{(n)}} - \left\lbrack \frac{\left( {P_{n} - P_{f_{n}} - {BP}} \right)*C_{1}}{\frac{\left( {P_{n + 1} - P_{n}} \right) - \left( {P_{f_{n + 1}} - P_{f_{n}}} \right)}{\left( {Z_{{TVD}{({n + 1})}} - Z_{{TVD}{(n)}}} \right)}} \right\rbrack}} & {{Equation}\mspace{14mu} 20.1} \\{{ETOFL} = {Z_{{TVD}{(n)}} - \left\lbrack \frac{\left( {P_{n} - P_{f_{n}} - {BP}} \right)*C_{1}}{MA\_ ISD} \right\rbrack}} & {{Equation}\mspace{14mu} 20.2}\end{matrix}$

where ETOFL represents the equivalent top of fluid level which isessentially equivalent to the fluid elevation in a well including afluid having a static density, P represents the measured pressure, P_(f)represents the frictional pressure loss, BP represents the surfaceannular applied back pressure, n represents a pressure sensor at somemeasured depth, and n+1 represents a pressure sensor at some deepermeasured depth.

Theoretical or Extrapolated Surface Annular Back Pressure

In MPD operations it may be useful to compute a theoretical orextrapolated surface annular back pressure (BP) from the measureddownhole annular pressures and to compare the computed values with theactual surface annular back pressure (SBP). Automated software routinesmay then be employed to adjust the actual applied BP so as to minimizeany differences to maintain a constant bottom hole pressure (BHP).

Equations 20.1 and 20.2 show that an increase in the interval density ata given BP results in an increase in ETOFL. This increase in intervaldensity may cause the theoretical back pressure in Equations 20.1 and20.2 to decrease and even go negative in some cases. In an event causinga sudden increase in the annular pressure measured by the lowermost pairof sensors (e.g., due to a restriction in the drill string above thesensors), the lowermost interval density remains substantially constant,ETOFL decreases, and the computed surface annular back pressure (SBP)increases. Since the theoretical BP depends on the interval from whichit is computed and the impact that various events have on the intervaldensity, interpretation of the theoretical BP is application dependantas described in more detail below with respect to Table 10. In generalinterpretation of the theoretical BP is used in combination with acomputed interval density in order to obtain the proper action foradjusting the actual surface back pressure.

The theoretical back pressure BP may be expressed mathematically, forexample, as follows:

$\begin{matrix}{{B\; P} = {{{- \left( Z_{n} \right)}*\left\lbrack \frac{\left( {P_{n + 1} - P_{n}} \right)}{\left( {Z_{{TVD}{({n + 1})}} - Z_{{TVD}{(n)}}} \right)} \right\rbrack} + P_{n}}} & {{Equation}\mspace{14mu} 21}\end{matrix}$

where BP represents the theoretical back pressure, P_(n) and P_(n+1)represent the measured pressures at sensors n and n+1, and Z_(TVD(n))and Z_(TVD(n+1)) represent the true vertical depths of sensors n andn+1.

Velocity and Acceleration of Interval Density Changes

It is often desirable to know the direction and degree of change in thecomputed interval specific gravities with time in order to determine ifthe system is tending towards stability or instability, and for example,tracking an inflow as it moves up the annulus. The rate of change of theinterval density may be represented mathematically, for example, asfollows:

$\begin{matrix}{{V\; I\; D} = {\frac{\mathbb{d}({ID})}{\mathbb{d}t} = \frac{\left( {{ID}_{t\; 2} - {ID}_{t\; 1}} \right)}{\left( {t_{2} - t_{1}} \right)}}} & {{Equation}\mspace{14mu} 22}\end{matrix}$

where VID represents the rate of change of the interval density withtime and ID_(t) represents one of the interval densities described aboveat time t.

A further derivative of the rate of change (i.e., an acceleration) mayalso be useful in determining the direction of the change and howquickly the interval density is changing in order to determine thenecessary reaction time for remedial action. The acceleration may alsohelp distinguish between gas kicks versus water or oil inflows. Intervaldensity acceleration may be expressed mathematically, for example, asfollows:

$\begin{matrix}{{AID} = {\frac{\mathbb{d}\left( {V\; I\; D} \right)}{\mathbb{d}t} = \frac{{VID}_{t\; 2} - {VID}_{t\; 1}}{t_{2} - t_{1}}}} & {{Equation}\mspace{14mu} 23}\end{matrix}$

where AID represents the rate of change of the velocity of the intervaldensity with time (i.e., the rate of change of the rate of change of theinterval density) and VID_(t) represents one of the velocities of theinterval densities at time t.

Interval Density Applications

In this section methodologies for interpreting the computed intervaldensities are presented along with several applications for usingcomputed interval densities to determine, diagnose, manage, and/orremedy various drilling events.

Interpretive Methodology

Table 1 summarizes the various interval densities described above in theINTERVAL DENSITY COMPUTATION METHODOLOGIES section and the physicaleffects that are included in each. The mathematical equations listedabove may be used to compute the various interval densities. Thecomputations may be performed in substantially real time while the wellis being drilled or subsequent to the drilling operation using recordedhistorical data. The disclosed embodiments are not limited in thisregard. The computed interval densities as well as their depth and timerelationships may be plotted on various crossplots or other displaysenabling the driller (or a computer software program) to recognize,differentiate, and take control of mitigating various situationsdiscussed later in this section. Moreover, use of the computed intervaldensities is not limited to drilling operations, but may also be usefulin various completion and production operations.

TABLE 1 Actual Actual Interval Actual Actual Modeled Modeled ActualModeled Annular Modeled Internal Modeled Density Pressure TemperaturePressure temperature Cuttings Cuttings Friction Annular FrictionInternal Computation Effects Effects Effects Effects Effects EffectsEffects Friction Effects Friction Comments MIF_ISD ● ● Internal frictioneffects are effectively removed MIF_ICD ● ● ● Measured while CirculatingEIF_ISD ● ● To be compared to MIF_ISD EIF_ICD ● ● ● To be compared toMIF_ICD MA_ISD ● ● ● Annular friction effects are effectively removed bytaking measurements when not circulating. MA_ISD_(mf) ● ● ● ● Annularfriction effects are effectively removed by modeling. MA_ICD ● ● ● ●Measured while Circulating EAF_ISD ● ● EA_ISD ● ● ● To be compared toMA_ISD EA_ICD ● ● ● ● To be compared to MA_ICD

EIF_ICD and EIF_ISD are the modeled (expected) internal intervalcirculating and static densities computed using the surface input mudproperties, including downhole pressure and temperature in the drillstring at the depth of interest. The expected quantities may be compareddirectly with the measured internal interval circulating and staticdensities MIF_ICD and MIF_ISD. MIF_ISD may be obtained by subtracting aninternal frictional pressure loss from the measured MIF_ICD or by directmeasurement. The frictional pressure losses may be obtained via modelingand/or measurements. For example, MIF_ICD may be measured directly bymeasuring MA_ISD when the mud pumps are turned off (e.g., when adding alength of drill pipe to the drill string). The difference betweenMIF_ICD measurements made while circulating and not circulating (whenthe pumps are on and off) may be considered to be a direct measurementof the internal frictional pressure losses (ΔP_Internal_(fric)).

The modeled EIF_ISD may be compared with MIF_ISD (which isMIF_ICD−ΔP_Internal_(fric) when circulating and MIF_ISD when notcirculating). An error minimization process (or a manual procedure) maybe used to adjust the hydraulic model parameters that account forpressure and temperature effects on the drilling fluid such that EIF_ISDequals MIF_ISD. A subsequent error minimization process may then beemployed to adjust the hydraulic model parameters that account forinternal frictional pressure losses such that EIF_ICD equals MIF_ICD(i.e., such that the modeled frictional pressure loss equals to themeasured frictional pressure loss ΔP_Internal_(fric)). Iterativeminimization processes may be utilized to provide for accurate results.The minimization processes may also be repeated at various flow ratesand the results stored in a look-up table for future reference.

The hydraulic model parameters obtained above for the pressure andtemperature effects on the input mud properties may be utilized in theannulus environment as well. The annular fluid properties as a functionof the annular pressure and temperature may be input to the hydraulicmodel to obtain a modeled (expected) annular fluid interval staticdensity EAF_ISD. This parameter represents the interval density of theannular fluid (without cuttings and friction effects) as a function ofannular pressure and temperature as a function of depth and time.Calibration and determination of the annular friction effects may beaccomplished in the same manner as described above for the internalfrictional effects. For these minimizations, EA_ISD, EA_ICD, MA_ISD andMA_ICD are computed as opposed to EIF_ISD, EIF_ICD, MIF_ISD and MIF_ISDas described in the preceding paragraph.

The modeled annular interval static density EA_ISD may be utilized asthe input mud properties with annular pressure and temperature andmodeled cuttings effects included. EA_ISD may be equal to MA_ISD whenthe generation and transport of cuttings in the annulus is properlymodeled and the modeled frictional pressure losses that are subtractedfrom MA_ICD are correct. An error minimization process may be utilizedto compute a cuttings density using appropriate values for frictionaltransport efficiency, ROP, porosity, and the density of the cuttingsfree fluid flowing in the annulus determined from the minimizationdescribed above for EAF_ISD. Changes in the computed cuttings density byinterval may indicate that cuttings are dropping out of suspension sincethe modeled cuttings density is constant with depth. A cuttingsmanagement process may track the loss of cuttings in the annulus andindicate the potential for undesirable drilling events such as pack-offswhile drilling or while reaming or pulling out of the hole.

Disclosed method embodiments may further utilize measurements of theactual flow into and out of each interval (e.g., as described above withrespect to Equation 8). Such measurements provide for a determination ofQ_(x) and may therefore be used to differentiate between inflow oroutflow effects versus incorrect cuttings modeling effects such as theassumed cuttings density. When flow in does not equal flow out,differences may be attributed to the quantity f_(x)·SG_(x) in Equation17 indicating flow in or out of the annulus in the interval in which thedifference occurs. In certain applications the interval densities maythen be used to compute the fractional volume and density of an inflowmaterial (e.g., using Equations 8-17). This process may be useful fordistinguishing between gas and salt water kicks, for example.

MA_ICD and EA_ICD may be equal when the various parameters discussedabove are modeled correctly. Differences between these two quantitiesmay also indicate undesirable drilling events as discussed above.Additionally, modeled frictional effects may depend on the boreholediameter. Using an LWD caliper, these effects can be properly accountedfor. However, with time the borehole wall may experience washout orenlargement, for example, due to drilling practices, shale stability, orother geomechanical effects. Differences in MA_ICD and EA_ICD may beused to detect and monitor borehole diameter changes. A minimizationprocess may also be used to determine the average borehole size withineach interval as a function of time.

The annular frictional losses also depend on the drill pipe rotationspeed (RPM) and fluid flow rate. Since these parameters may change withtime, the annular frictional effects can also be time dependant and maybe accounted for during drilling.

Effects of Pressure and Temperature on Fluid Densities

The fluid or mud being pumped into the well while drilling may beaffected by the pressure and temperature changes it undergoes as ittravels down the drill pipe and back up the annulus. For example,pressure and temperature changes cause corresponding changes to thedensity of the fluid. These changes may be measured using theaforementioned ASM measurements and may enable the relationship betweenfluid density and pressure and temperature to be quantified and/ormodeled which in turn enables other effects such as cuttings loading andfriction to be determined.

Internal ASM pressures, temperatures, and computed interval densitiesand temperature gradients may be used with a hydraulic model tocalibrate the model parameters. The hydraulic model may then be used topredict the effects at any other point in the system as a function ofdepth and time. Annular measurements may be used in the same mannerunder non-drilling conditions (i.e., when there are no cuttings in theannular fluid). When the hydraulic model parameters are well defined andpredictable for a particular drilling fluid, and in cases where either ameasured temperature or measured pressure is not available, thehydraulic model may be used to predict the missing measurement.

FIG. 4 depicts modeled oil based mud (OBM) density as a function ofpressure and temperature. As indicated at 402 and 404, the density ofthe mud increases with decreasing temperature 402 and increasingpressure 404. Under circulating conditions in which the OBM temperatureremains somewhat constant (i.e., does not increase significantly withdepth), OBM density increases with depth (and therefore pressure) asindicated at 406. Under non-circulating conditions in which the OBMtemperature increases significantly with depth, the temperature effectcan overwhelm the pressure effect (i.e., the fluid density can decreasewith increasing depth as indicated at 408).

FIGS. 5A and 5B depict one example of a log including computed intervaldensities obtained during an ASM while drilling operation. Table 2summarizes the relative locations of the annular pressure measurementswhen the drill bit was located at a measured depth of 17,000 feet. Thelowermost annular pressure measurement was made in a SchlumbergerarcVISION® tool deployed in the BHA. This pressure measurement islabeled “APRS” in track 2 (at 502). The drill string further includedfirst and second ASM annular pressure sensors labeled “1231” and “1244”in track 2. The 1244 sensor was located about 1259 feet (in measureddepth) and 787 feet (in true vertical depth) above the BHA annularpressure measurement. The 1231 sensor was located about 5777 feet (inmeasured depth) and 5603 feet (in true vertical depth) above the 1244sensor. A surface measurement SPPA was located about 9934 feet above the1231 sensor.

TABLE 2 Bit Sensor offset Sensor Sensor Sensor Depth from bit MDInclination TVD Surface 17000 17000 0 0 0 ASM 1231 17000 7066 9934 0.29932 ASM 1244 17000 1289 15711 62 15535 APRS from 17000 30 16970 6416322 arcVISION tool in BHA

Table 3 summarizes the parameters depicted on FIGS. 5A and 5B. Many ofthese parameters are described above in the INTERVAL DENSITY COMPUTATIONMETHODOLOGIES section and are further described in more detail belowwith respect to the present example.

TABLE 3 Track Curve name Definition 1 WDP Status 0 = down 1 = up 2 APRSarcVISION annular Pressure 1231AnnularPressure Annular Pressure from ASMsensor #1231 1244AnnularPressure Annular Pressure from ASM sensor #1244SPPA Surface Stand Pipe Pressure 3 MA_ED_001 ECD calculation from APRSmeasurement using TVD of sensor MA_ED_003 ECD calculation from ASM 1244measurement using TVD of sensor MA_ED_009 ECD calculation from ASM 1231measurement using TVD of sensor 4 MA_IED_003_001 Interval densitycalculation between ASM sensor 1231 and surface annular pressure.MA_IED_009_003 Interval density calculation between ASM sensor 1244 andASM 1231 sensor annular pressure. MA_IED_999_009 Interval densitycalculation between ASM sensor 1244 and surface sensor. 5 MA_TOM_003_001ETOFL estimate calculated from APRS and ASM sensor 1244 pressures.MA_TOM_009_003 ETOFL estimate calculated from ASM 1244 and 1231 sensorspressures. MA_TOM_009_001 ETOFL estimate calculated from APRS and ASMsensor 1231 pressures. 6 MA_TOM_003_001 Calculated surface back pressureusing APRS and 1244 sensor measurements. MA_TOM_009_003 Calculatedsurface back pressure using 1244 and 1231 sensor measurements.

With continued reference to FIGS. 5A and 5B, track 7 (depicted at 504)includes the densities and interval densities computed between theaforementioned pressure sensors in the BHA and the drill string. Theannular mud density is computed for each individual sensor and labeledMA_EC (measured annular equivalent circulating density). MA_ED_001corresponds to the equivalent density for the APRS pressure measurement,MA_ED_003 corresponds to the 1244 pressure measurement, and MA_ED_009corresponds to the 1231 pressure measurement. These parameters tend tobe insensitive to heterogeneities in the local mud density asillustrated in this example by the fact that the values at each sensorare substantially identical and overlay one another on the plot. Whilenot depicted on FIGS. 5A and 5B, the computed equivalent density foreach of the sensors has a value about equal to the density of the baseOBM (about 7.9 ppg or 0.95 g/cm³). When the pumps are shut down at thesimulated connection (from 14:35 to 15:05 in track 1), these densitiesdrop as expected due to the lack of annular friction losses.

The computed interval densities are also shown in track 4 (506) and arelabeled as MA_IED_003_001 (the interval density between the APRS and1244 sensors), MA_IED_003_009 (the interval density between the 1244 and1231 sensors), and MA_IED_999_009 (the interval density between the 1244ASM sensor and the surface annular pressure sensor). When the pumps areshut down at the connection, the interval densities drop due to theelimination of annular friction losses. The interval densities areessentially the aforementioned quantities MA_ICD when circulating andMA_ISD when not circulating. In this particular example, the intervaldensities also closely represent the EAF_ISD since the rate ofpenetration (ROP) was low and there were long periods of circulationbetween drilling events, implying there were little to no cuttingssuspended in the annular fluid.

The uppermost interval density (MA_IED_999_009) is approximately equalto the computed equivalent densities shown in track 3 (at 8 ppg). Asdepicted in track 4, the interval densities decrease significantly withincreasing depth, with MA_IED_003_009 being about equal to 7.6 ppg andMA_IED_003_001 being about equal to 7.3 ppg. The decreasing intervaldensities are likely due to increasing temperatures lower in thewellbore. Absent such temperature effects, one would expect the densityof a compressible fluid such as an OBM to increase with increasingdepth. However, as shown on FIG. 4, the increasing temperature of thedrilling fluid with increasing depth can result in a decreasing density.This may be observed directly using the interval densities disclosedherein (as depicted on FIGS. 5A and 5B).

With still further reference to FIGS. 5A and 5B, tracks 5 and 6 (shownat 508 and 510) depict equivalent top of fluid (ETOFL) and computed backpressure. In track 5, the top of fluid levels are labeled MA_TOM_003_001(the interval between the APRS and 1244 sensors), MA_TOM_003_009 (theinterval between the 1244 and 1231 sensors), and MA_TOM_009_001 (theinterval between the APRS and 1231 sensors). In track 6, the backpressures are labeled MA_BP_003_001 (the interval between the APRS and1244 sensors) and MA_BP_003_009 (the interval between the 1244 and 1231sensors). As depicted, the computed back pressures have positive values.The annular choke pressure may be set to a value equal to the valuecalculated for the lowermost pair of sensors (MA_BP_003_001) in track 6in order to maintain a constant bottom hole annular pressure whendrilling a narrow mud weight window. Upon resuming circulation, thelowermost sensor (APWD) measures the full annular friction pressureabove the sensor (in addition to the static pressure) while thosesensors located further uphole sense diminishing frictional losses. Theresulting interval densities are therefore larger than the correspondinginterval static densities.

In well drilling operations, the borehole temperature commonly increaseswith increasing depth. Under circulating (and drilling) conditions, thetemperature of the drilling fluid is generally not a strong function ofdepth (due to the mixing of the fluid and transport back to thesurface). When circulation stops, the temperature typically increaseswith time and any particular depth until a steady-state temperature isreached. As a result, the density of the drilling fluid may also beexpected to decrease with time after circulation ceases. These timedependent changes in density may also be observed using theaforementioned interval densities.

The ASM pressure and temperature measurements and their relationship tofluid density may be further utilized in refining and/or calibratingconventional hydraulic models. For example, the measurements may beutilized to determine the coefficients in the conventional API-13Dequations:ρ_(base)=(a ₁ +b ₁ P+c ₁ P ²)+(a ₂ +b ₂ P+c ₂ P ²)T  Equation 24ρ_(brine)=(a ₃ +b ₃ P+c ₃ P ²)+(a ₄ +b ₄ P+c ₄ P ²)T  Equation 25

where ρ_(base) represents the density of the base drilling OBM,ρ_(brine) represents the density of the brine, P represents pressure, Trepresents temperature, and a, b, and c represent fitting coefficients.Table 4 includes sample “book” values for various conventional oiland/or brine solutions as well as fitting statistics and range ofvalidity.

TABLE 4 Calcium Chloride Internal 19.3 wt % Diesel Mineral Oil OlefinParaffin Pressure Coefficients a₁ (lb_(m)/gal) 9.9952 7.3183 6.99126.8358 6.9692 b₁ (lb_(m)/gal/psi) 1.77E−05 5.27E−05 2.25E−05 2.23E−05  3.35E−05 c₁ (lb_(m)/gal/psi²)   6E−11   −8E−10   −1E−10   −2E−10  −5E−10 Temperature Coefficients a₂ (lb_(m)/gal/° F.) −2.75E−03  −3.15E−03   −3.28E−03   −3.39E−03   −3.46E−03 b₂ (lb_(m)/gal/psi/° F.)3.49E−08 7.46E−08 1.17E−07 1.12E−07 −1.64E−08 c₂ (lb_(m)/gal/psi²/° F.)  −9E−13   −1E−12   −3E−12   −2E−12     2E−13 Fitting Statistics forModeled Data Avg. Error % 0.135 0.237 0.166 0.194 0.214 r² coefficient0.998 0.997 0.998 0.998 0.999 Range of Validity Max. Applied Pressure(psi) 20,300 20,000 20,300 24,000 14,500 Min. Temperature (° F.) 77 4077 56.4 68 Max. Temperature (° F.) 392 400 392 392 302

It may be advantageous in certain applications to adjust these “book”values according to in-situ conditions. Since the oil to water ratio isknown (it is commonly controlled at the surface), Equations 24 and 25may be combined into a single equation having six coefficients, forexample as follows:ρ_(mud)=(i ₁ +j ₁ P+k ₁ P ²)+(i ₂ +j ₂ P+k ₂ P ²)T  Equation 26

where ρ_(mud) represents the density of the drilling fluid (thecombination of base and brine) and i, j, and k represent thecoefficients. This density may be measured in-situ, for example, usingthe aforementioned interval density computations where the pressure andtemperature values represent an average value for the interval.

A drill string including six ASM pressure and temperature sensors, forexample, may enable the six coefficients to be computed. For example,six interval densities may be calculated using the corresponding sixpressure and temperature measurements to obtain six equations having sixunknowns (the six coefficients). Values for the coefficients may then bedetermined using conventional root finding algorithms. It will beunderstood that the necessary number of intervals may be reduced, forexample, via using minimization techniques or using interval densitiescomputed at multiple times (or multiple depths) provided that thepressure and temperature measurements are sufficiently different.

Alternatively, Equations 24 and 25 may be combined into a singleequation having twelve coefficients, for example as follows:

$\begin{matrix}{\mspace{79mu}{{\rho_{mud} = {{MIF\_ ISD} = \frac{{V_{base} \cdot \rho_{base}} + {V_{brine} \cdot \rho_{brine}}}{V_{mix}}}};}} & {{Equation}\mspace{14mu} 27} \\{{MIF\_ ISD} = {{V_{base}\left\lfloor {\left( {a_{1} + {b_{1}P} + {c_{1}P^{2}}} \right) + {\left( {a_{2} + {b_{2}P} + {c_{2}P^{2}}} \right)T}} \right\rfloor} + {\left( {1 - V_{base}} \right)\left\lbrack {\left( {a_{3} + {b_{3}P} + {c_{3}P^{2}}} \right) + {\left( {a_{4} + {b_{4}P} + {c_{4}P^{2}}} \right)T}} \right\rbrack}}} & {{Equation}\mspace{14mu} 28}\end{matrix}$

where V_(base) and V_(brine) represent the volume fractions of base andbrine. The coefficients in Equations 27 and 28 may be obtained by making12 independent interval density measurements, for example, at twodistinct locations using the drill string described above having six ASMpressure and temperature sensors.

In another alternative embodiment, values for the brine coefficients(a₃, b₃, c₃ and a₄, b₄, c₄ in Equations 25 and 28) may be assumed andthe six base coefficients evaluated, for example, using at least sixindependent interval density measurements.

In the foregoing embodiments, the coefficients may be determined usingeither internal interval density measurements or annular intervaldensity measurements. Internal interval density measurements may bepreferred due to the lack of cuttings in the interior of the drillstring, however, annular measurements may also be utilized when thecuttings are accounted for using one or more of the aforementionedtechniques.

Cuttings Transport Efficiencies and Formation Characterization

ASM pressure and temperature measurements may be utilized to detectchanges in cuttings densities and transport efficiencies and maytherefore further be utilized in characterizing the lithology of theformation being drilled. As described above with respect to Equations8-17, the ASM pressure measurements may be used to determine constituentdensities of various materials in the drilling fluid. In operations inwhich there is no annular inflow or outflow (i.e., when Q_(x) and f_(x)are approximately equal to zero), the cuttings density may be readilydetermined using EA_ISD and MA_ISD.

FIGS. 6, 7, and 8 depict a hypothetical example of a well drillingoperation in which a change in formation lithology is encountered thatresults in a reduced cuttings density. In each of FIGS. 6, 7, and 8,track 2 (shown at 604) schematically depicts the lithology beingdrilled, for example, as determined by a computed cuttings density and adimensionless torque. The drill pipe and drill bit are shown at 622 and624, while the outline of the borehole is shown at 626. Cuttings arefurther depicted at 628 as being transported to the surface in thedrilling fluid moving upward through the annulus. The depicted drillstring includes four along string pressure and temperature sensors 630A,630B, 630C, and 630D and a surface sensor 632. It will be understoodthat the disclosed embodiments are not limited to any particular numberof ASM sensors.

Track 1 depicts (at 602) MIF_ISD and EIF_ISD, the former of which iscomputed from MIF_ICD by subtracting the modeled and/or measuredinternal drill pipe frictional effects on the flowing mud. EIF_ISDrepresents the input mud density properties corrected for the effects ofthe internal drill pipe measured and/or modeled pressures andtemperatures using a suitable hydraulic modeling program. The necessaryhydraulic modeling parameters for the pressure and temperature effectsmay be determined by matching EIF_ISD to MIF_ISD over the intervalswhere MIF_ISD computations are available.

Track 3 includes (at 606) the computed annular interval densities,EAF_ISD, MA_ISD, EA_ISD, MA_ICD, and EA_ICD. EAF_ISD represents thedensity of the cuttings free input mud flowing up the annulus correctedfor the measured annulus pressures and temperatures using the samehydraulic modeling parameters determined for the internal mud. Themodeled cuttings load is added to EAF_ISD to obtain EA_ISD. The measuredinterval static density MA_ISD is equal to the measured intervalcirculating density MA_ICD less the annular frictional losses when thecuttings volume, density, and transport and the frictional flowparameters are properly modeled. A minimization program may be utilizedin the modeling as described above in to achieve this as describedabove.

Track 4 depicts (at 608) the computed cuttings density. Other parametersare shown on Tracks 5-8 and discussed in more below with regards toother examples. It will be understood in FIGS. 6, 7, and 8 that when twoparameters (e.g., represented by dashed and solid curves) are equal toone another, they are shown with a slight separation (approximately acurve width) in order to make both curves visible. Such representationis merely convenience and not meant to be limiting.

Time differentials of the measured interval static and circulatingdensities MA_ISD and MA_ICD are shown in track 5 at 610. Equivalent topof fluid ETOFL for the static and circulating fluid are shown in track 6at 612. Calculated annular back pressure BP for the static andcirculating fluid are shown in track 7 at 614 and the measured annulusstatic and circulating pressures P are shown in track 8 and 616.

FIG. 6 depicts the hypothetical drilling operation at time t₁=0. Asshown in track 3 (at 606), the measured and expected annulus static andcirculating densities are equal to one another (i.e., MA_ISD isapproximately equal to EA_ISD and MA_ICD is approximately equal toEA_ICD). The computed cuttings density shown in track 4 is constant withdepth indicating that the time required for the cuttings to reach thesurface is less than the time taken to drill the present formationlayer. The volume fraction of the cuttings f_(cuttings) decreasestowards the top of the borehole (as shown schematically on track 2) andmay be due, for example, to the rate of penetration, formation porosity,and/or cuttings transport effects as a function of time. These variablesmay be accounted for in the minimization process. The quantityf_(cuttings) may also be shown in the log if desired.

FIG. 7 depicts the hypothetical drilling operation shown in FIG. 6 attime t₂=t₁+Δt and includes the same tracks as described above. Asdepicted on track 2, the drill bit has penetrated a new formation havinga lower density, thereby resulting in cuttings 629 having a lowerdensity than the previously generated cuttings 628. As a direct resultof the reduced cuttings density, MA_ISD falls below EA_ISD and MA_ICDfalls below EA_ICD in the lower most interval (as depicted at 702 and704 in track 3).

It will be understood that a change in cuttings density may beidentified by signatures other than those discussed above with respectto FIG. 7. Tables 5A and 5B list the expected signatures that resultfrom such a change in the cuttings density in the annulus (typically asa result of drilling a new formation before the minimization process hascomputed a new cuttings density value). Table 5A lists expectedsignatures when drilling a formation having a lower density while Table5B lists expected signatures when drilling a formation having a higherdensity.

TABLE 5A Parameter Changes with Time Changes with Depth Q_(x) Q_(x) = 0;No change Q_(x) = 0; No change F_(x) may change may change SG_(cuttings)Computed value will decrease NA MA_ISD vs. EA_ISD MA_ISD < EA_ISD;Decrease in MA_ISD moves uphole MA_ISD decreasing with time as lightercuttings move uphole in until interval contains only the the annulus newlighter cuttings MA_ICD vs. EA_ICD MA_ICD < EA_ICD; Decrease in MA_ICDmoves uphole MA_ICD decreasing with time as lighter cuttings move upholein until interval contains only the the annulus new lighter cuttings.ETOFL ETOFL decreases with time ETOFL is lower over intervals overaffected interval. having lighter cuttings. Calculated Annular BPincreases with time over BP is higher over intervals having Surface BPaffected interval. lighter cuttings. ASM Pressure Slight decrease withtime over Slight decrease over intervals having affected interval.lighter cuttings. ASM Temperature No change No change

TABLE 5B Parameter Changes with Time Changes with Depth Q_(x) Q_(x) = 0;No change Q_(x) = 0; No change F_(x) may change may change SG_(cuttings)Computed value will increase NA MA_ISD vs. EA_ISD MA_ISD > EA_ISDIncrease in MA_ISD moves uphole MA_ISD increasing with time as heaviercuttings move uphole in until interval contains only the the annulus newheavier cuttings. MA_ICD vs. EA_ICD MA_ICD > EA_ICD Increase in MA_ICDmoves uphole MA_ICD increasing with time as heavier cuttings move upholein until interval contains only the the annulus new heavier cuttings.ETOFL ETOFL increases with time ETOFL is higher over intervals overaffected interval. having heavier cuttings. Calculated Annular BPdecreases with time over BP is lower over intervals having Surface BPaffected interval. heavier cuttings. ASM Pressure Slight increase withtime over Slight increase over intervals having affected interval.heavier cuttings. ASM Temperature No change No change

FIG. 8 depicts the hypothetical drilling operation shown on FIG. 6 attime t₃=t₂+Δt and includes the same tracks as described above. WhenQ_(x)=0 a minimization program may be used to directly determine thedensity of the cuttings. This new cuttings density is depicted in track4 at 802 and indicates a reduced cuttings density as expected. The newcuttings density may also be utilized to compute new expected intervalcirculating and static densities EA_ICD and EA_ISD, which areapproximately equal to the corresponding measured interval densitiesMA_ICD and MA_ISD as shown in track 3 at 804 and 806.

The cuttings density SG_(cuttings) may be used, for example, to identifythe lithology of the formation being drilled (e.g., sandstone,limestone, dolomite, shale, tar, salt, etc.). For example, quartzsandstone has a density of about 2.65, calcium carbonate limestone has adensity of about 2.71, calcium magnesium carbonate dolomite has adensity of about SG of 2.85, mixed mineral shale formations have anaverage density in the range from about 2.6 to about 2.7, halite saltshave a density of about 2.17, tar layers have a density in the rangefrom about 0.8 to about 1.1, and anhydrite has a density of about 2.97.Knowledge of the cuttings velocity (or velocities) with time, enablescuttings depths to be assigned, which in turn may enable a lithology log(e.g., as depicted in track 2) to be constructed. In the exampledepicted on FIGS. 6-8, the cuttings density in the interval beingdrilled is less than that of the previous interval which further aids inthe identification of the formation lithology.

Those of ordinary skill in the art will readily appreciate thatformation bulk density is a widely used petrophysics parameter. Thisparameter is commonly used for applications ranging from overburdencalculations, geomechanical modeling, synthetic seismograms, andformation porosity determination. The formation bulk density isgenerally a function of the lithology (or mineral content of theformation) and the fluid type and volume in the formation. In drillingoperations in which the drilling process destroys the formationporosity, the computed cuttings density may be used as the mineraldensity (formation matrix density) to compute the porosity from aborehole geophysical measurement of bulk density.

Tar Mat Identification

Tar zones (also referred to in the art as tar mats) are a common threatin drilling operations and can at times represent a serious risk to adrilling operation. Since tar is difficult to identify in seismic maps,avoidance can be challenging and often relies primarily on localexperience. Moreover common utilized logging while drilling (LWD)technologies, such as gamma ray and resistivity logging measurements,are not always capable of identifying tar zones. As such a drillingoperator sometimes does not realize that a tar zone has been intercepteduntil the annulus is full of tar. This can result in a pack-offsituation and a stuck BHA. The ASM pressure and temperature measurementsand the interval densities disclosed herein may be used to quicklyidentify and mitigate intercepted tar zones.

The disclosed interval densities may be utilized to identify tar in theannulus via computing the interval cuttings density as described abovewith respect to FIGS. 6-8 and Tables 5A and 5B. The presence of tar inthe annulus may be identified by a decrease in the lowermost intervaldensity. This decrease may be modeled as a corresponding decrease in thecomputed cuttings density. Tar mats tend to cause a significant decreasein the interval density for at least two reasons. First, the density ofthe tar is significantly less than that of the rock formations commonlydrilled (e.g., in a range from about 0.8 to about 1.1 as compared to arange from about 2 to about 3 for the drilled rock as described above).Second, the tar mats generally include a high volume fraction of tar(many tar mats are non porous layers that are made up nearly 100% tar)such that the volume fraction of tar in the local annular interval isalso high.

Early identification of tar mats enables the drilling operator tomitigate the influx of tar into wellbore. Such mitigation may includeany number of techniques, for example, including, the use of managedpressure to artificially boost the constraining pressure or backpressure in the annulus to keep additional tar from sloughing into theborehole, moving the pipe up above the point of the tar mat withoutcirculating, then introducing a heavier weight mud into the borehole(called spotting a pill), side tracking around the tar, treating the tarwith various chemical additives, and isolating the tar via the use ofvarious types of casing. The disclosed embodiments are, or course, notlimited to any particular mitigating action.

Borehole Washout

Due to various geomechanical and/or drilling practices the borehole canbecome enlarged with time during a drilling operation. Such boreholeenlargement can be detrimental for several reasons. For example, anenlarged borehole can reduce the velocity of cuttings moving up throughthe annulus thereby enhancing the possibility of cuttings dropping outof suspension and packing off the borehole. Enlarged boreholes alsorequire larger volumes of cement during casing operations.

FIGS. 6, 9, and 10 depict a hypothetical example of another welldrilling operation in which a portion of the borehole becomes enlargedduring the drilling operation (FIGS. 9 and 10 depict the enlargement).This example again uses FIG. 6 to depict the hypothetical drillingoperation at time t₁=0. As described above, the measured and expectedinterval densities are substantially equal to one another along thelength of the wellbore (i.e., MA_ISD=EA_ISD and MA_ICD=EA_ICD as shownin track 3) indicating that the cuttings volume, cuttings density,cuttings transport and fractional volume, and the annular frictionaleffects have been properly modeled.

FIG. 9 depicts the hypothetical drilling operation at time t₂=t₁+Δt andincludes the same tracks as described above with respect to FIG. 6. Awashout zone having an increased diameter is depicted at 902 in track 2.In track 3 at 904, MA_ICD has decreased and is less than EA_ICD in thewashout interval, however, MA_ISD remains substantially constant and isabout equal to EA_ISD as shown at 906. The enlarged borehole causes theannular frictional pressures to decrease in the washout interval therebyreducing the measured interval circulating densities, but not theexpected interval densities that are computed using a model that makesuse of LWD caliper measurements or the bit size when the interval wasdrilled. The measured and expected interval static densities remainsubstantially equal since the washout is at a constant depth and sincethe cuttings are not dropping out of suspension in this example.MA_ISD_(mf) which is computed by subtracting a modeled annular frictionfrom MA_ICD also decreases in the washout interval as shown at 908. Intrack 5 at 910, the derivative of MA_ICD is negative indicating a dropin MA_ICD with time as the borehole washes out (becomes enlarged).

FIG. 10 depicts the hypothetical drilling operation at time t₃=t₂+Δt. Aminimization process has been instructed to compute a new boreholediameter such that the expected annulus frictional pressures are reducedand match the measured interval circulating density. As depicted intrack 3 at 1002, MA_ICD and EA_ICD are now substantially equal in thewashout interval (as a result of the minimization process creating alarger borehole diameter). This new diameter may be stored as a functionof time for plotting and analysis against drilling practices andparameters and time dependant formation strength determinations tofurther enhance the understanding of the formation strength and toacknowledge and prevent the practice of detrimental drilling practicesin the future. Moreover, the borehole diameter computed at the end ofthe drilling process may be used to calculate the volume of cementneeded in the post-drilling casing operation.

It will be understood that a change in borehole diameter (e.g., causedby a washout) may cause corresponding changes in certain of thedisclosed parameters other than those described above with respect toFIGS. 9-10. Table 6 lists the expected changes caused by a boreholewashout or enlargement.

TABLE 6 Parameter Changes with Time Changes with Depth Q_(x) Q_(x) = 0;No change Q_(x) = 0; No change f_(x) May decrease and can change withMay decrease and can change as time other intervals washoutSG_(cuttings) No change No change MA_ISD vs. EA_ISD MA_ISD constant withtime and MA_ISD = EA_ISD at same depth equal to EA_ISD during washout.interval, not moving uphole unless MA_ISD > MA_ISD_(mf) other intervalsare enlarging. MA_ICD vs. EA_ICD MA_ICD decreasing with time and MA_ICDless than EA_ICD over less than EA_ICD during washout. affected depthinterval, not moving uphole unless other intervals are enlarging.Circulating ETOFL Decreases as washout enlarges with Decreases aswashout enlarges, time. Static ETOFL not changing. Remains at fixeddepth. Circulating Calculated Increases as washout enlarges withIncreases as washout enlarges, Surface annular BP time. Static BP notchanging. Remains at fixed depth. ASM Pressure Slight decrease ofcirculating Slight decrease, can change as other pressure duringenlargement. intervals washout. ASM Temperature Slight increase withtime due to Slight increase due to flow velocity flow velocity decrease.decrease at that depthBorehole Pack-Off

As used in the art, a pack-off describes a situation in which theborehole diameter has been reduced creating a “choke” to fluid flowingup the annulus. Such a reduction may be caused, for example, by a largevolume of cuttings that have dropped out of suspension in the annulus orsloughing of the borehole wall into the annulus. With insufficientannular fluid velocity, mud viscosity, or in a highly inclined borehole,the cuttings may accumulate at some depth in the well and cause arestriction (the pack-off). Depending upon the severity of the pack-off,the pressure may increase to undesirable levels deeper in the well andmay even cause the formations to fracture if remedial action is notperformed in a timely manner. The pack-off can also result in lostcirculation which in turn can cause a loss of hydrostatic head and apossible inflow or even a kick from a permeable formation. A severepack-off can even also result in a stuck BHA if sufficient cuttings areallowed to accumulate around the drill string.

FIGS. 11, 12, and 13 depict a hypothetical example of a well drillingoperation in which borehole cuttings drop out of suspension and form apack-off. Track 2 of FIG. 11 includes an enlargement at 1102 asdescribed above with respect to FIGS. 9 and 10. In FIGS. 12 and 13 apack-off is depicted just below the enlargement at 1202. FIGS. 11-13display the same tracks as described above in FIGS. 6-8. In thisexample, FIG. 11 depicts the hypothetical drilling operation at timet₁=0 (after the formation of the washout). It will be understood thatthe disclosed embodiments are not limited by the depiction of a washout.In track 3 of FIG. 11, the measured and expected interval densities aresubstantially equal to one another along the length of the wellbore(i.e., MA_ISD=EA_ISD and MA_ICD=EA_ICD) indicating that the cuttingsvolume, cuttings density, cuttings transport and fractional volume, andthe annular frictional effects have been properly modeled.

The pack-off is depicted schematically in track 3 (at 1202) in FIGS. 12and 13. The restriction causes the annular circulating pressures furtherdown in the well to increase as shown at 1204 in track 8 of FIG. 12. Thecirculating pressure above the restriction may also decrease slightly asdepicted at 1206 if the flow rate is significantly reduced above thepack-off. Conventional annular pressure measurements by themselves mayat times be used to identify the pack-off by monitoring annular pressurechanges with time and depth. The disclosed interval densities may alsobe utilized to identify a pack-off and tend to provide a more definitivesignature. For example, as depicted on FIG. 12, the interval densitiesthat span the pack-off tend to increase while the interval densitiesabove and below this span tend to remain unchanged. Within the pack-offspan, the measured interval densities MA_ISD_(mf) and MA_ICD increasesignificantly over the corresponding expected (modeled) intervaldensities EA_ISD and EA_ICD as depicted at 1208 and 1210. When the pumpsare brought down and the actual static density is measured, MA_ISD_(mf)is also observed to be larger than the measured interval static densityMA_ISD. Moreover, MA_ISD may be approximately equal to (or possiblyslightly greater than) EA_ISD as shown at 1212 depending on the mass ofaccumulated cuttings. Q_(x) is also observed to be approximately equalto zero as indicated at 1214 in FIG. 12. These observed signatures tendto be uniquely attributable to pack-offs (or other annular restrictions)with the added characteristic that the aforementioned interval densitydifferences remain at a fixed depth (since the pack-off itself remainsat fixed depth).

FIG. 13 is similar to FIG. 12, but depicts an alternative methodologyfor computing the interval densities. In particular, each of theintervals used in FIG. 13 extends from the depth of the ASM sensor tothe surface (instead of the interval between adjacent sensors asdepicted on FIG. 12). In FIG. 13, each of the measured intervalcirculating densities below the pack-off is greater than thecorresponding expected interval circulating density as depicted at 1302and 1304. The calculated ETOFL and BP are zero by definition when usingthis calculation technique as shown in tracks 6 and 7. In themethodology shown on FIG. 13, the interval densities from the pack-offlocation to the drill bit increase. This may advantageously make thevisual impact of the event more noticeable in certain displayconfigurations and may further enable the axial location of the pack-offto be estimated.

It will be understood that the development of a pack-off or arestriction may cause corresponding changes in certain of the disclosedparameters other than those described above with respect to FIGS. 12 and13. Table 7 lists the expected changes caused by a pack-off or annularrestriction.

TABLE 7 Parameter Changes with Time Changes with Depth Q_(x) Q_(x) = 0;No change Q_(x) = 0; No change F_(x) No change No change SG_(cuttings)No change No change MA_ISD_(mf) vs. MA_ISD_(mf) > MA_ISD MA_ISD_(mf) >EA_ISD MA_ISD Increases with time as pack-off Over pack-off depthinterval only Develops MA_ISD vs. EA_ISD MA_ISD ≈ EA_ISD MA_ISD ≈ EA_ISDOver pack-off depth interval MA_ICD vs. EA_ICD MA_ICD > EA_ICD MA_ICD >EA_ICD Increases with time as pack-off Over pack-off depth interval onlyDevelops Estimated Top of Circulating ETOFL increasing Circulating ETOFLincreasing Fluid across event, slightly decreasing across event,slightly decreasing below event, and no change above below event, and nochange above event, all changing as pack-off event. Pack-off intervalhas the develops. Static ETOFL not largest ETOFL. Static ETOFL notaffected if pack-off interval is short. affected if pack-off interval isshort. Calculated Annular Circulating BP decreasing across CirculatingBP decreasing across Surface BP event, slightly increasing below event,slightly increasing below event, and no change above event, event, andno change above event. all changing as pack-off develops. Pack-offinterval has the lowest BP. ASM Pressure Circulating pressures increaseCirculating pressures increase below the pack-off, no change below thepack-off, no change above pack-off, increases as pack- above pack-off.off develops. ASM Temperature Slight increase below pack-off with Slightincrease below pack-off, time, decrease above pack-off decrease abovepack-off

The identification of the pack-off by observing annular pressures andinterval densities may be automated such that the signature shown inFIG. 12 (e.g., MA_ISD>EA_ISD and MA_ICD>EA_ICD with the differences notchanging with time) triggers an alarm that alerts the drilling operator.The automation routine may further reduce the circulation rate to reducethe pressure buildup below the pack-off. The drilling operator may theninitiate a sequence of steps designed to break-up or dislodge thepack-off (e.g., working the drill string up and down in the boreholewhile rotating). It will be understood that the disclosed embodimentsare not limited in these regards.

Inflow to the Borehole Annulus

As is known to those of ordinary skill in the art, formation fluids tendto flow into the wellbore during drilling when the formation has ahigher pore pressure than the mud pressure at the formation depth. Suchinflow events can occur further up the borehole if the mud column isallowed to drop below the surface, for example, when tripping the drillpipe out of the borehole. Swab events can also contribute to an inflow.Formation fluids, such as gas, oil, or connate water, generally exhibita lower density than the drilling mud. Any inflow therefore tends tofurther reduce the hydrostatic head, allowing the inflow rate toincrease until the wellbore can no longer be controlled. Timelymitigation therefore requires early recognition of the inflow event. ASMpressure and temperature measurements and the disclosed intervaldensities may be used to identify inflow events soon after they begin.

FIGS. 14, 15, 16, and 17 depict a hypothetical example of a welldrilling operation including a formation fluid inflow event (alsoreferred to as a kick). Track 2 of FIG. 14 depicts the drill bitpenetrating a new formation 1402. In FIGS. 15-17 formation fluid influxis depicted at 1502 in track 2. FIGS. 14-17 display the same tracks asdescribed above in FIGS. 6-8. In this example, FIG. 14 depicts thehypothetical drilling operation at time t₁=0 (after penetratingformation 1402 but prior to the fluid influx event shown on FIGS.15-17). It will be understood that the disclosed embodiments are notlimited by the depiction of the fluid inflow coming for the bottom ofthe well. Inflow may occur substantially anywhere along the length ofthe borehole as is known to those of ordinary skill in the art. In track3 of FIG. 14, the measured and expected interval densities aresubstantially equal to one another along the length of the wellbore(i.e., MA_ISD=EA_ISD and MA_ICD=EA_ICD) indicating that the cuttingsvolume, cuttings density, cuttings transport and fractional volume, andthe annular frictional effects have been properly modeled. Moreover, asshown at 1404, Q_(x) is approximately equal to zero indicating noinflow.

FIG. 15 depicts the hypothetical drilling operation at time t₂=t₁+Δt.The inflow event has started as depicted at 1502 of track 2 causingQ_(x) to be greater than zero as depicted at 1508. The parameter Q_(x)may be estimated via a surface measurement of the difference in flowrate between the flow out of the annulus and the flow into the drillstring (a differential flow volume). Equations 8-17 described above maybe used to estimate or more accurately determine Q_(x). In someinstances a simple difference between the flow rate out of the annulusand the flow rate into the drill string may be suitable to estimate avalue of Q_(x). More accurate values of Q_(x) may be obtained by takinginto account Q_(cuttings) generated from the drilling operation asdisclosed in Equations 8-17. In normal drilling operations, Q_(cuttings)may be in a range, for example, from about 1 to about 5 percent of thedrilling fluid flow rate. An inflow event (e.g., a kick) may be resultin Q_(x) being in a range, for example, from about 5 to about 100percent or more of the drilling fluid flow rate.

With continued reference to FIG. 15, the measured interval static andcirculating densities MA_ISD and MA_ICD decrease below the correspondingexpected values EA_ISD and EA_ICD as shown at 1504 and 1506 in track 3.Since Q_(x)≠0 the program logic retains the most recent value ofSG_(cuttings) as indicated at 1510 (and via comparison of track 4 inFIGS. 14 and 15).

FIG. 16 depicts the hypothetical drilling operation at time t₃=t₂+Δt. Aminimization process is used instead to compute a value for the densityof the inflow material SG_(x) as indicated at 1610 in track 4 of FIG. 16(e.g., using Equations 8-17). The computed density of the inflowmaterial SG_(x) may then be utilized to estimate the type of fluidcoming into the annulus. For example, a gas influx may have a density ofless than about 0.6, an oil influx may have a density in a range fromabout 0.6 to about 0.8, and a connate water influx may have a density ofabout 1 to about 1.2. After assigning a value for SG_(x) the measuredinterval static and circulating densities MS_ISD and MS_ICD are againapproximately equal to the expected values ES_ISD and ES_ICD as shown at1602 and 1604.

FIG. 17 depicts the hypothetical drilling operation at time t₄=t₃+Δt. Asthe inflow rises or is circulated up the annulus as shown at 1702 intrack 2 of FIG. 17, the computed SG_(x) moves up the annulus as well asshown at 1710 in track 4. This further illustrates the signaturedifferences between an inflow and a pack-off or a borehole enlargementwhere the pressure disturbance remains at a constant depth. Moreover thederivative of the interval densities (shown at 1612 and 1712 of FIGS. 16and 17) indicate how rapidly the inflow is moving up the annulus,thereby facilitating the planning of the particular control methodologyused to control the well.

With continued reference to FIGS. 14-17, the Equivalent top of fluidlevel ETOFL becomes negative in the annular intervals having the inflowmaterial (e.g., as indicated at 1512 in track 6 of FIG. 15).Furthermore, the calculated surface annular back pressure BP becomespositive in the annular intervals having the inflow material (e.g., asindicated at 1514 in track 7 of FIG. 15). As the inflow material movesup the wellbore, the ETOFL decreases (or goes negative) and BP increases(or goes positive) in progressively higher intervals in the borehole.

FIG. 18 depicts one example of a visual display illustrating inflow as afunction of time and depth. Depth is shown on the vertical axisincreasing in the downward direction. Time is shown on the horizontalaxis increasing to the right. Interval density values are plotted ascontours (for example using pseudo-color enhancement with warmer colorsrepresenting lower interval density values—but using grey scale contoursin the depicted example in which a darker shade represents lowerinterval density values). The black regions are below the bit in thedepicted example and therefore include no data. The left screen at timet₁ represents a snapshot of a time interval in which drilling isprogressing. A lighter interval density is shown to be appearing at thelowermost interval on the right at 1802. The subsequent screensrepresent subsequent times t₂, t₃, and t₄ in which the a kick ofcomparatively low density fluid is moving up the annulus with time (thetime progression is indicated at 1804, 1806, and 1808).

It will be understood that the development of an inflow (or kick) maycause corresponding changes in certain of the disclosed parameters otherthan those described above with respect to FIGS. 14-17. Table 8 liststhe expected changes caused by an inflow before SG_(x) and Q_(x) havebeen computed (e.g., via the aforementioned minimization processes) andadjusted the expected annulus interval densities EA_ISD and EA_ICD.

TABLE 8 Parameter Changes with Time Changes with Depth Q_(x) Q_(x) > 0;May change with time Q_(x) > 0 F_(x) No change No change SG_(cuttings)No change No change MA_ISD vs. EA_ISD MA_ISD < EA_ISD MA_ISD < EA_ISDDifference increases with time if Moving up the annulus with time ifinflow continues the inflow continues MA_ICD vs. EA_ICD MA_ICD < EA_ICDMA_ICD < EA_ICD Difference increases with time if Moving up the annuluswith time if inflow continues the inflow continues Equivalent top offluid ETOFL is negative in the intervals ETOFL is negative in theintervals containing the inflow and containing the inflow and inflowdecreasing with time if inflow effect will move up the annulus continueswith time Calculated Surface BP is positive and increasing with BP ispositive in the intervals annular BP time if inflow continues containingthe inflow, and inflow effect will move up the annulus with time ASMPressure Decreases with time if inflow Decrease in the intervalscontaining Continues the inflow, and inflow effect will move up theannulus with time ASM Temperature Depends on the influx temperature,Highest rate of change at influx influx type, and pressure if there aredepth, changes migrate uphole with Joule-Thomson effects. Changes theinflux fluid. with time if influx rate changes.

During formation fluid sampling operations, formation fluid may bepumped (or released) into the annulus. For example, formation fluid isoften pumped into the annulus for a period of time prior to sampling theformation fluid to ensure that only virgin fluid is sampled (i.e., thatthe sampled fluid is not contaminated with drilling fluid or cuttings).Up to one barrel or more of formation fluid may be released into theannulus for each sample acquired. The density of the annular fluid maybe monitored while sampling using the interval density techniquesdescribes herein. Moreover, after the samples are acquired, theformation fluid may be circulated to the surface and released through anannular choke. The interval densities may also be used to monitor theupward movement of the formation fluid through the annulus, therebypotentially saving considerable rig time.

When an inflow event (e.g., a kick) is encountered, a drilling operatormay elect to circulate through an annular choke while heavy mud ispumped downhole. The disclosed interval densities may continue to bemeasured and computed and used to determine when the bottom hole densityand pressure is sufficient to stop the inflow. For example, a measuredbottom hole pressure may be used to drive a choke to keep the pressurewithin a desired range while pumping the heavy mud.

Outflow from the Borehole Annulus

Annular fluids may flow into the formation as it is drilled when theformation has a lower pore pressure than the drilling fluid pressure atthat depth. Such an outflow may happen at the bit or further up theborehole if the drilling fluid pressure is allowed to increase above theformation pressure. In some operations, an outflow reduces thehydrostatic head thereby causing the outflow rate to decrease until thewellbore stabilizes. Such outflow events may be thought of asself-mitigating. However, in other operations, the reduced hydrostatichead caused by the outflow may trigger an inflow (or kick) in anotherformation (e.g., at another location in the borehole). As describedabove, inflow events can lead to highly dangerous and uncontrollablewell conditions. Timely mitigation requires early recognition of theproblem, and in keeping with the purposes of this section, timelyrecognition of the outflow event. ASM pressure and temperaturemeasurements and the disclosed interval densities may be used toidentify outflow events soon after they begin.

FIGS. 14, 19, and 20 depict a hypothetical example of a well drillingoperation including a drilling fluid outflow event. Track 2 of FIG. 14depicts the drill bit penetrating a new formation 1402 as describedabove with respect to FIGS. 14-17. In FIGS. 19 and 20, outflow ofdrilling fluid into the formation is depicted at 1902 in track 2. FIGS.14, 19, and 20 display the same tracks as described above in FIGS. 6-8.In this example, FIG. 14 depicts the hypothetical drilling operation attime t₁=0 (after penetrating formation 1402 but prior to the fluidoutflow event shown on FIGS. 19 and 20). It will be understood that thedisclosed embodiments are not limited by the depiction of the fluidexiting the bottom of the well. Outflow may occur substantially anywherealong the length of the borehole as is known to those of ordinary skillin the art. In track 3 of FIG. 14, the measured and expected intervaldensities are substantially equal to one another along the length of thewellbore (i.e., MA_ISD=EA_ISD and MA_ICD=EA_ICD) indicating that thecuttings volume, cuttings density, cuttings transport and fractionalvolume, and the annular frictional effects have been properly modeled.Moreover, as shown at 1404, Q_(x) is approximately equal to zeroindicating no inflow or outflow.

With continued reference to FIG. 14, the circulating and static top offluid levels ETOFL are shown on track 6. These values may be computedfrom the measured interval static densities MA_ISD (e.g., according toEquation 20). As depicted, ETOFL from the surface to the first pressuresensor is zero. The ETOFL values tend to vary downhole, however the netsum or average is approximately zero. The calculated surface annularback pressure BP anti-correlates with ETOFL (as shown on track 7) andagain averages approximately zero at the t₁=0 conditions.

FIG. 19 depicts the hypothetical drilling operation shown at timet₂=t₁+Δt. The outflow event has started as depicted at 1902 of track 2causing Q_(x) to be less than zero as depicted at 1908. The parameterQ_(x) may be obtained as described above with respect to FIG. 15. In thedepicted example, the drilling fluid level in the annulus has droppedbelow the surface due to the outflow as shown at 1904 in track 2 (e.g.,during static wellbore conditions). The measured static and circulatingpressures are less than the pre-outflow values as depicted at 1912 and1914 in track 8. The interval densities MA_ICD and MA_ISD have decreasedin the interval containing the liquid level and any intervals above thatone as shown at 1906 and 1907 of track 3. These values may (or may not)drop below EAF_ISD depending on the liquid level, cuttings loading andannular frictional effects. The derivatives of the interval circulatingand static densities are negative within and above the intervalcontaining the liquid level and zero in the intervals below the intervalcontaining the liquid level as shown at 1916 and 1918 of track 5.

With continued reference to FIG. 19, the ETOFL values have increased atall intervals containing a full column of drilling fluid as shown at1922 such that the sum or average has become positive. FIG. 19 depicts ascenario in which the fluid level is above the uppermost ASM pressuresensor 630D. In this example, the interval between the surface anduppermost pressure has a zero-valued ETOFL by definition. The intervaldirectly below the interval containing the liquid level may be taken tohave a high quality ETOFL and BP values. The calculated average surfaceannular BP is negative. The average value represents the initial amountof reduction in the actual BP for the MPD surface equipment. As the BPis reduced, gas or nitrogen may come out of solution thereby reducingthe density of the annular fluid in a positive feedback condition. If noBP is being applied, the bottom hole pressure (BHP) of the lowermostsensor extrapolated to total depth represents the formation porepressure and maximum BHP for drilling ahead.

FIG. 20 is similar to FIG. 19, but depicts a scenario in which thedrilling fluid level has dropped below the first ASM (note that fluidlevel 1904 is below uppermost ASM sensor 2002). In this scenario theinterval including the fluid level now has a non-zero ETOFL and BP asshown at 2004 and 2006 in tracks 6 and 7. Moreover, the intervaldensities MA_ISD and MA_ICD are near zero in the uppermost interval asshown at 2008 in track 3 since this interval contains no fluid. TheETOFL and BP values may again be obtained from the first interval belowthe fluid level.

It will be understood that while the annular fluid level may drop duringa lost circulation event, the internal drill-pipe fluid level may or maynot coincide with the annular fluid level due to differing pressuresabove and below both fluid levels. This condition is sometimes referredto as in the art as “U-tubing”. Internal pressure measurements may beused to determine the fluid levels in the interior of the drill-pipe inan analogous manner to the method described above for the annular fluidlevel. Moreover, in extreme lost circulation events, the fluid level inthe annulus may drop during circulation while drilling fluid is beingpumped down the interior of the drill string.

It will be understood that the development of an outflow may causecorresponding changes in certain of the disclosed parameters other thanthose described above with respect to FIGS. 14, 19, and 20. Table 9lists the expected changes caused by an outflow. It will be understoodthat the minimization may not be required to compute the new expectedinterval densities EA_ISD and EA_ICD.

TABLE 9 Parameter Changes with Time Changes with Depth Q_(x) Q_(x) < 0;May change with time Q_(x) < 0; May change with depth F_(x) No change Nochange SG_(cuttings) No change No change MA_ISD vs. EA_ISD MA_ISD <EA_ISD MA_ISD < EA_ISD Difference changes until liquid Moving down theannulus with level stabilizes. MA_ISD decreases time until liquid levelstabilizes. with time over the affected intervals MA_ISD drops below orclose to which are the intervals above and EAF_ISD in interval havingliquid including the fluid level. level. MA_ISD and MA_ICD in intervalsbelow liquid level not be affected. MA_ICD vs. EA_ICD MA_ICD < EA_ICDMA_ICD < EA_ICD Difference changes until liquid Moving down the annuluswith level stabilizes. MA_ICD decreases time until liquid levelstabilizes. with time over the affected intervals MA_ICD drops below orclose to which are the intervals above and EAF_ISD in interval havingliquid including the fluid level. level. MA_ICD closely approachesMA_ISD in interval containing the fluid level and equals MA_ISD inintervals above the fluid level in which non-liquids are present.Equivalent top of fluid Both static and circulating ETOFL Both staticand circulating ETOFL increase with time in each interval is positive inthe intervals below below the interval containing the and including theliquid level. fluid level until fluid level Moves down until liquidlevel stabilizes. Average of all intervals stabilizes. Interval belowfluid level is positive. has representative ETOFL. Average of allintervals is positive. Calculated Surface Both static and circulating BPBoth static and circulating BP are annular BP decrease with time in eachinterval negative in the intervals below and below the intervalcontaining the including the liquid level. Moves fluid level until fluidlevel down until liquid level stabilizes. stabilizes. Average of allintervals is negative. ASM Pressure Decreases in all sensors. DecreasesDecreases in all sensors until liquid with time if outflow continues.level stabilizes. Amount of decrease will be the same for all sensorsbelow the fluid level for incompressible fluids. ASM TemperatureIncreases in all intervals due to lack May increase in affectedintervals of circulation. Increases with time. due to lack ofcirculation.

In response to an outflow event a drilling operator often shuts in thewell, stops pumping, and closes the annular choke until pressuresstabilize. The interval densities may be utilized to determine theliquid level of the drilling fluid while the ASM and APWD measurementsmay be used to obtain the BHP when the liquid level stabilizes. This BHPthen becomes the maximum BHP that should be applied during the futuredrilling operations. When drilling restarts, the flow rate may bereduced and/or nitrogen may be injected into the input flow stream toreduce the density of the drilling fluid sufficiently so that the BHPremains below the maximum value. The average calculated annular BP orany one of the interval calculated BP or the downhole measured annuluspressures may be used in an automatic choke control. As disclosedherein, the choke position may be controlled in time intervals by anelectro-mechanical server to reduce the BP by the amount calculateduntil the system stabilizes.

FIG. 21 (including FIGS. 21A and 21B) depicts an example log from a welldrilling operation in which drilling fluid was lost during the drillingoperation. The depicted log is time stamped in track 1 (FIG. 21A). Thelowermost annular pressure measurement was made in a SchlumbergerarcVISION® tool deployed in the BHA. This pressure measurement islabeled APRS in track 3. The drill string further included first andsecond ASM annular pressure sensors labeled 1231 and 1244 in track 3.Density values based on a single sensor measurement are plotted in track4. MA_ED_001 corresponds to the APRS pressure measurement, MA_ED_003corresponds to the 1244 pressure measurement, and MA_ED_009 correspondsto the 1231 pressure measurement. Interval densities are plotted intrack 5 (FIG. 21B). MA_IED_003_001 corresponds to the interval betweenthe APRS and 1244 pressure measurements, MA_IED_003_009 corresponds tothe interval between the 1244 and 1231 pressure measurements, andMA_IED_999_009 corresponds to the interval between the 1231 pressuremeasurement and the surface. Equivalent top of fluid values for each ofthe aforementioned intervals are plotted in track 6.

In the depicted example, downhole dynamics sensors detected a highdegree of stick/slip in a measured depth range from about 5152 to about5179 meters. A viscous pill was pumped on 14-December 16:00 one whilethe back pressure was kept at 350 psi. This was observed to stabilizethe whole and drilling continued at a controlled rate of penetration to5199 meters. On 15 December 07:20 the applied torque increased from 8000to about 12,700 foot pounds and partial fluid losses were thought tooccur based on bit level observations. At 07:42 pressures were observedto drop significantly in response to a lost circulation event and a lossof hydrostatic head. At the APRS sensor, the pressure dropped from about7500 to about 6800 psi as indicated at 2102. The interval densitybetween the APRS and 1244 pressure sensors also dropped from about 8.5to about 5 ppg as indicated at 2104, while the other two intervaldensities remain approximately unchanged (dropping from about 8.5 toabout 8 ppg) as indicated at 2106. Moreover the ETOFL of the lowermostinterval the first spiked to a positive value before dropping to about−10,000 feet as indicated by the wraparound at 2108. These resultsstrongly indicate a lost circulation event in the lowermost interval,likely at the bit. Drilling and circulation was subsequently suspended.

FIGS. 22A and 22B depict schematic depth vs. pressure plots illustratingETOFL changes that may result from lost circulation events. In FIG. 22Athe lost circulation event occurs at (or near) the bit. Prior to theevent (at time t=0), the depth vs. pressure curve is approximatelylinear as indicated at 2202. At time t=1, the lost circulation eventcauses a pressure drop at the lowermost sensor ASM1 which may result inan increasing ETOFL (above the surface) in the lowermost interval(between ASM1 and ASM2) as indicated by the increased slope at 2204. Astime progresses and the ETOFL may decrease significantly as indicated at2206 (and 2108 of FIG. 21B).

FIG. 22B depicts a schematic depth vs. pressure plot for a lostcirculation event that occurs above the bit (between ASM2 and ASM4 inthis example). Prior to the event, the depth vs. pressure curve isapproximately linear as indicated at 2212. As circulation is lost themeasured pressures drop at sensors ASM3 and ASM4. This may result in anincreased ETOFL (above the surface) in the interval between sensors ASM3and ASM4 as indicated at 2214 and a decreased ETOFL between sensors ASM2and ASM3 as indicated at 2216. This signature strongly suggests a lostcirculation event above the bit (e.g. nearby to ASM3 in FIG. 22B).

FIG. 23 (including FIGS. 23A and 23B) depicts an example log from thewell drilling operation depicted in FIG. 21 taken about one day later(the morning of 16-December). The same tracks and data flow aredepicted. After drilling was discontinued (as described above withrespect to FIG. 21), the BHA was pulled uphole to 5093 meters measureddepth without circulation. An attempt was made to regain circulation ata low flow rate without success. After pulling the BHA back into thecasing for a period of time, then tripping back to bottom, drillingfluid was again pumped into the well. The aforementioned intervaldensities and equivalent top of fluid were monitored while filling. TheETOFL can be seen to be rising with filling at 2302. Pumping wassuspended at 06:51 and fluid level shots were performed using anEchometer. The Echometer detected a fluid depth of 2038 feet which iscomparable to the average ETOFL of 2000 feet shown at 2304 on FIG. 23B.

Managed Pressure Drilling Choke Adjustments

During managed pressure drilling (MPD) operations, the surface annularback pressure (SBP) is maintained such that the bottom hole pressure(BHP) remains in a predefined small range in order to prevent both lostcirculation and kicks or wellbore stability issues. For example, as themud pumps are brought down, the surface annular back pressure may beincreased in order to compensate for the loss of annular friction and isalso adjusted (up or down) to account for possible phase changes whenusing aerated (or nitrogenated) drilling fluid. Automated feedbackcontrol is desirable in order to make the adjustment more timely andaccurate. Moreover, automatic control may be further desirable in theevent of drilling condition changes (e.g., a kick or change in cuttingsdensity). The back pressure calculations disclosed herein may providefor such automated feedback.

FIG. 24 (including FIGS. 24A and 24B) depicts an example log from thesame well drilling operation as was depicted in FIG. 21. Tracks 1through 7 are identical to FIGS. 21 and 23. Track 8 is added andincludes a computed interval back pressure BP using Equation 21.MA_BP_003_001 corresponds to the BP computed for the interval betweenthe APRS and 1244 pressure measurements while MA_IED_003_009 correspondsto BP computed for the interval between the 1244 and 1231 pressuremeasurements. OPT_LINE_1 plots the actual SBP.

In FIGS. 24, 24A and 24B, logging data is shown that corresponds to atime interval prior to making a connection (December 13 23:10-23:30) inwhich the pumps were shut down, but the wired drill pipe remainedconnected. Annular back pressure was being applied; however there was nonitrogen injection. The average back pressure during prior drilling(e.g., at 22:20) was about 350 psi. When shutting the pumps down at23:10, back pressure was increased by 275 psi to 625 psi to compensatefor the loss of annular friction. The downhole pressure measurements atthe APRS, 1231, and 1244 sensors are seen to increase by about 100-150psi above the drilling value at 2402, 2403, and 2404 in track 3 (FIG.24A). The APRS pressure measurement is reproduced in track 7 at 2406using the same resolution as the SBP (FIG. 24B).

In this operation the goal was to minimize the pressure overshoot andreduce the pressure to the drilling value. The overshoot was reduced bylowering the back pressure over the following 10 minutes (from 23:10 to23:20) as indicated at 2408. In this operation, a back pressure of about525-550 psi appears optimal for compensating for the loss of annularfriction losses. Therefore, the annular pressure losses due to frictionwere about 175 psi, rather than the 275 psi originally assumed. Suchcalibration of the back pressure may improve stability and eliminateinflow issues at connections.

Track 8 displays the computed BP. These computed back pressures indicatethe efficiency at which the SBP is being transmitted to the drillingfluid in the annulus at any particular interval. The computed BP may becompared directly in a control loop to obtain a desirable SBP, forexample, via adjusting the SBP such that the SBP and computed BP areapproximately equal. Since a constant BHP is desirable, theMA_BP_003_001 data may be used directly in the control loop. In FIGS.24A and 24B there are several intervals in which swab effects areobserved, e.g., between 23:22 and 23:27. In such instances, the computedBP is higher than the actual SBP implying that SBP should be increasedwhich would in turn decrease the computed BP. The aforementioned controlloop may be configured, for example, to incrementally increase SBP untilSBP is approximately equal to the computed BP. Such a loop tends to beinherently stable since these quantities generally move in oppositedirections (e.g., increasing SBP decreases BP and decreasing SBPincreases BP). When surge effects take place (e.g., between 22:50 and22:55), the computed BP is lower than the actual SBP. The SBP shouldtherefore be lowered.

The above described methodology for controlling back pressure duringmanaged pressure drilling operations may be advantageously highly stablesince the computed back pressure (from Equation 21) is sensitive to thetransmission efficiency of the applied SBP to the annular fluid.

In maintaining a desired BHP during MPD operations, the input flow ratemay be adjusted, the mud weight may be adjusted, the volume of injectednitrogen varied, or the BP may be adjusted. In many cases two or more ofthese parameters may be adjusted substantially simultaneously. Moreover,the average calculated annular BP or any one of the interval calculatedBP or the measured downhole measured annulus pressure may be used in anautomatic choke control methodology. The choke position may becontrolled, for example, in incremental steps by an electro-mechanicaldevice until the system stabilizes and BP and SBP are substantiallyequal as described above.

Table 10 lists the direction of change for the theoretical BPcalculation across the depth intervals while certain other drillingevents take place (other than compensating for annular friction lossesas described above). These events are listed in column 1. Column 2 liststhe desired change in the surface BP during MPD operations in order tocounter-act the event down-hole and to maintain a substantially constantBHP (or to maintain the BHP within a safe mud weight window).

TABLE 10 Desired Surface BP Theoretical or Calculated BP EventAdjustment BP across event BP below event Drilling lighter IncreaseSurface BP Circulating and static NA Cuttings BP will increase atlowermost sensor pair Drilling heavier Decrease Surface BP Circulatingand static NA Cuttings BP will decrease at lowermost sensor pair WashoutIncrease Surface BP Circulating BP will No Change increase acrosswashout. Static BP constant Pack-off Decrease Surface BP Circulating BPwill Circulating BP will decrease across pack- slightly increase belowthe off. Static BP constant pack-off Kick Increase Surface BPCirculating and static Circulating and static BP BP will increase willslightly decrease across kick interval below kick interval if applicableLost Circulation Decrease Surface BP Circulating and static Circulatingand static BP BP will slightly will decrease in all decrease acrossfluid intervals below fluid level level Mud Rheology or Decrease SurfaceBP If interval density NA property changes since BHP will be increases,circulating resulting in increasing and static BP will interval densitydecrease increases Mud Rheology or Increase Surface BP If intervaldensity NA property changes since BHP will be decreases, circulatingresulting in decreasing and static BP will interval density increasedecreases In-situ Monitoring of Drilling Fluid Health

As described above, the internal ASM pressures and temperatures may beused to measure the input mud density and temperature profiles. Theinternal ASM measurements may be further used to compute hydraulicmodeling parameters that are in turn used to predict subsequent pressureand temperature effects on the annular fluid as it moves up the annulus.When changing the mud weight or other properties such as the viscosityduring a viscous sweep, it may be beneficial to know where the viscousmud (or pill) is in the system. When the mud becomes uniform within thesystem, drilling can resume.

A circulating time or bottoms up time may be used to determine the depthfrom which the cuttings collected at the surface have come. Many timesthe driller will circulate “bottoms up” before POOH (Pull Out Of Hole).This is estimated using an estimated borehole diameter and volume whichcan be in error. Since the time needed to clean the borehole of allcuttings is not well defined, a safety factor of 1.5 to 2 is commonlyused, meaning that circulation time is increased by these factors toinsure a clean hole before POOH.

The interval densities and annular friction tend not to change with timeonce the mud is homogeneous. Non-changing interval densities maytherefore be used to determine when the mud density is homogeneouswithin the borehole volumes. When the annulus is free of cuttings, theannular interval densities tend to reflect the density of the input mudcorrected for pressure and temperature effects. Circulation can then bestopped in order to POOH. Either or both of Equations 22 and 23 may beused to determine when the mud system is homogeneous and other drillingoperations have resumed.

Production Analysis

Obtaining production in wells, especially lateral wells, is oftencomplicated by conveyance issues. In a lateral well, deployment ofdownhole tools through standard gravity descent may not be possible. Toovercome this difficulty, the tools may be either pushed or pulled intothe well by means of drill pipe assisted logging, tubing conveyance,tractored, propelled with a swab cup, or some other means. Theaccumulation of debris while conveying various production tools into thewell can be particularly problematic in horizontal or near horizontalwells. Moreover excessive rig time is often required for conveyingconventional wireline (WL) tools into horizontal wells such that WLtools are sometimes not used.

Wireline conveyed production analysis tools often include numerousmeasurement sensors deployed at various depths in the wellbore. Suchmeasurement sensors may alternatively be deployed using wired drill pipeconveyance. The use of WDP enables substantially identical sensors to bedeployed in the same configuration and at multiple depths in thewellbore. Sensor deployment may be accomplished via tripping the WDPinto the bore hole. The surface pressure may be adjusted such thatformation fluids flow into the wellbore and up the interior of the drillpip where they may be vented through a surface choke or routed toproduction facilities. The along string pressure and temperaturemeasurements as well as the computed interval densities and temperaturegradients may then be used to gauge the type and rate of fluid flow fromthe various intervals. Additionally, by controlling the uphole pressure,the effect of the pressure variability on the fluid properties down-holecan be assessed—such as phase changes, flow rate changes, liquid holdupchanges, and the like.

Cuttings Transport Management

Adequate transport of cuttings from the drill bit to the surface isnecessary in order to prevent various drilling problems such as frictioncaused by the accumulation of the cuttings, generation of a pack-offaround the BHA or other locations on the drill string, and stuck drillpipe. Increased friction due increased cuttings volume or barite sag inthe drilling fluid can slow the removal of the cuttings and result inone or more of the above problems. Cuttings transport issues, if notproperly identified and mitigated, can quickly spiral out of control,for example, from increased friction, to a pack-off, to a stuck drillpipe.

In high angle wells, for example including horizontal and nearhorizontal wells, there is an increased tendency for cuttings to dropout of suspension. This can occur for at least two reasons, includingthe non-uniform annular flow profile with stagnation increasing towardsthe bottom of the borehole and the action of gravity in a perpendiculardirection to the flow velocity. Having only a short distance to fallinto the stagnation flow profile at the bottom of the bore hole, theaforementioned cutting transport problems can therefore manifest quicklyin high angle wells.

Various factors such as drill string rotation rate, drilling fluid flowrate, and periodic BHA and drill-pipe axial movements help to keep thecuttings bed stirred up and in suspension. However, at the time of thisdisclosure there is no known definitive down-hole measurement availableto measure the degree of success of these practices at specific depthintervals. Drilling personnel often wait to determine whether or nottargeted cuttings appear at the shale shakers approximately (e.g., 20-90minutes after penetration of the particular formation). Current practicemay also make use of single sensor BHA measurements from which drillingpersonnel look for increases in overall annular density with time todetect cuttings buildups. However, such a buildup may also be due todrilling denser rock with a high rate of penetration or to pack-offslocated above the BHA. It is commonly assumed that a decrease in annulardensity with time corresponds to better hole cleaning and cuttingstransport. In reality, cuttings dropping out of solution can give thesame signature. In contrast, the ASM pressure and temperaturemeasurements, computed interval densities, and their derivatives may beused to distinguish cuttings drop-out from other effects and locate theaffected depth intervals.

FIGS. 25 and 26 depict a hypothetical example of a well drillingoperation in which borehole cuttings drop out of suspension in adeviated borehole. Track 2 of FIG. 25 includes an enlargement at 2502 asdescribed above with respect to FIGS. 9 and 10. FIGS. 25 and 26 displaythe same tracks as described above in FIGS. 6-8. In this example, FIG.25 depicts the hypothetical drilling operation at time t₁=0 (after theformation of the washout but prior to cuttings dropping out ofsuspension). It will be understood that the disclosed embodiments arenot limited by the depiction of a washout. In track 3 of FIG. 25, themeasured and expected interval densities are substantially equal to oneanother along the length of the wellbore (i.e., MA_ISD=EA_ISD andMA_ICD=EA_ICD) indicating that the cuttings volume, cuttings density,cuttings transport and fractional volume, and the annular frictionaleffects have been properly modeled.

FIG. 26 depicts the hypothetical drilling operation at time t₂=t₁+Δt atwhich cuttings are dropping out of suspension. The dropped cuttings aredepicted schematically in track 2 (at 2602) in FIG. 26. As the cuttingsmove uphole from the bit, the cuttings density remains approximatelyconstant and may be tracked as a function of time and depth (e.g., afterSQ_(cuttings) stabilizes). When cuttings drop out of suspension,SQ_(cuttings) may decrease significantly (e.g., by about 10 to about 50percent).

An automated routine may be utilized to identify and quantify theseverity of a cuttings transport issue (e.g., dropped cuttings from theannular volume) as a function of time and depth prior to running theaforementioned minimization routine. When cuttings are dropping out ofsuspension, MA_ISD decreases below EA_ISD and approaches (or issubstantially equal to) EAF_ISD (as can be seen by comparing FIGS. 25and 26 at 2504 and 2604). MA_ICD also decreases below EA_ICD as depictedat 2606 of FIG. 26. The Equivalent top of fluid ETOFL may also decreasewhile the annular back pressure BP increases as depicted at 2608 and2610.

While the interval density changes tend to mimic those of a kicksignature and/or a lost circulation signature, cuttings transport issuescan be readily identified by noting that Q_(x)=0 in FIGS. 25 and 26.This distinguishes cuttings transport from inflow or outflow events.Note also that the routine holds SQ_(cuttings) constant as depicted at2612. In the event that SQ_(cuttings) is mistakenly computed instead ofbeing held constant by the program, the value of SG_(cuttings) may dropa value approximately equal to the mud density whereas during a kick(especially a gas kick), SG_(cuttings) drop to a value below the muddensity.

It will be understood that cuttings transport issues, especially ininclined wells, may cause corresponding changes in certain of thedisclosed parameters other than those described above with respect toFIGS. 25 and 26. Table 11 lists certain changes caused by cuttingsdropping out of suspension in the annulus. These changes are observedbefore a minimization routine has computed new interval density valuesand adjusted the expected annulus (EA) quantities accordingly.

TABLE 11 Parameter Changes with Time Changes with Depth Q_(x) Q_(x) = 0Q_(x) = 0 F_(x) May change May change SG_(cuttings) NA NA MA_ISD vs.EA_ISD MA_ISD equals EA_ISD until The ISD signatures tend to be cuttingsdrop-out occurs at affected at particular intervals where which timeMA_ISD drops cuttings drop-out is most probable, below EA_ISD andapproaches e.g., at 40-65 degree inclination. The the mud density.Differences depth intervals creating drop-out increase with time untildriller tend not to change with time. takes remedial action. Unlike aMA_ISD < MA_ISD_(mf). washout where MA_ISD remains constant and MA_ICDis affected. MA_ICD vs. EA_ICD MA_ICD and EA_ICD tend to Same signaturesas ISD curves. mimic the ISD signatures, although the effect may belarger or smaller depending on the drop out volume and the net effect onannular friction. Equivalent top of fluid ETOFL decreases with timeETOFL decreases over intervals over the affected intervals as wherecuttings are dropping out. cuttings drop out. Slight increase belowaffected intervals. Calculated Surface BP increases with time as BPincreases with time as cuttings annular BP cuttings drop out. drop out.Slight decrease below affected intervals. ASM Pressure Slight decreaseSlight decrease ASM Temperature No expected change No expected change

A driller may elect to respond to cuttings transport issues, such ascuttings falling out of suspension in the annulus, using a number ofmitigating techniques. For example, a drilling operator may elect to (i)increase the rotation rate of the drill string to promote turbulentmixing of the annular fluid, (ii) increase the drilling fluid flow rate,(iii) reduce the rate of penetration (e.g., via reducing weight on bit),or even (iv) replace the drill bit with a less aggressive bit or a bithaving a different nozzle configuration. Other BHA components may alsobe replaced so as to change the pressure drop between the surface andthe drill bit. The disclosed embodiments are not limited in any of theseregards.

Internal and External Temperature Gradients

Internal and annular temperature measurements made as a function ofdepth and time may be used to compute various temperature gradients inthe borehole. For example, internal and external (annular) temperaturegradients may be determined along the length of the drill string (as afunction of measured depth). Moreover, radial gradients through thedrill string between internal and external temperature measurements maybe determined. These temperature gradients may be utilized to evaluatevarious drill string and tool related conditions as well as variousformation related conditions.

In one embodiment, temperature gradients may be computed as a functionof both time and depth along the drill string to predict when theborehole temperature in the BHA may exceed rated tool temperatures.These measurements may be made in both circulating and staticconditions. In a high temperature formation the temperature of theborehole may increase with both time and depth during static conditions.Therefore, measured temperature gradients may enable the determinationof a time at which rated tool temperatures are exceeded. For example,LWD formation fluid sampling operations are generally carried out duringstatic conditions. The aforementioned temperature gradients may enable amaximum time-on-station to be determined during which the samplingoperation would need to be completed. Circulation may then be resumed soas to cool the BHA.

In another embodiment internal and external measurements may be used tomodel a radial heat transfer coefficient of the drill string or downholetool. Such modeling may further include a third temperature measurementto be made between the internal and external fluids (e.g., in aninternal circuit board). The use of three temperature measurements mayenable non-linear heat transfer effects to be evaluated. Suchmeasurements may be made during circulating and/or static conditions.These temperature measurements may be included in a model to predictdrill string temperatures for numerous drilling conditions. For example,temperature gradients may be evaluated at multiple drill string rotationrates (e.g., 50 rpm, 100 rpm, and 200 rpm) and at multiple drillingfluid flow rates (e.g., 300 gpm, 500 gpm, and 800 gpm). This may enablethe effects of various drilling parameters, including drill stringrotation rate and drilling fluid flow rate, in mitigating hightemperature drilling situations.

Developing a heat transfer model, for example, as described in thepreceding paragraph may further enable the measured temperatures to beused to calculate a static formation temperature. Obtaining the staticformation temperature may be highly valuable in that it is related tonumerous parameters of interest including formation heat transfercapacity which is in turn related to the fluid and lithology content ofthe formation which is still further related to the porosity,hydrocarbon saturation, and pore pressure. Determination of the staticformation temperature may further enable circulating and static boreholetemperatures to be predicted long before completing the well. Phasechanges may also be identified. Moreover knowledge of the staticformation temperature may enable staging plans to be refined whiletripping into hot wells.

Although numerous methods for computing and utilizing wellbore intervaldensities and certain advantages thereof have been described in detail,it should be understood that various changes, substitutions andalternations can be made herein without departing from the spirit andscope of the disclosure as defined by the appended claims.

What is claimed is:
 1. A method for computing a theoretical surfaceannular back pressure in a subterranean wellbore during a managedpressure drilling operation, the method comprising: (a) deploying a toolstring in the wellbore, the tool string including first and secondsubsurface longitudinally spaced pressure sensors deployed atcorresponding first and second measured depths in the wellbore; (b)causing the first and second pressure sensors to acquire first andsecond annular drilling fluid pressure measurements at the first andsecond measured depths; and (c) causing a processor to process the firstand second pressure measurements to compute the theoretical surfaceannular back pressure for a wellbore interval between the first andsecond measured depths, wherein the theoretical surface annular backpressure is computed according to the following equation:${B\; P} = {{{- \left( Z_{{TVD}{(1)}} \right)}*\left\lbrack \frac{\left( {P_{2} - P_{1}} \right)}{\left( {Z_{{TVD}{(2)}} - Z_{{TVD}{(1)}}} \right)} \right\rbrack} + P_{1}}$wherein BP represents the theoretical surface annular back pressure, P₁represents the first pressure measurement, P₂ represents the secondpressure measurement, and Z_(TVD(1)) and Z_(TVD(2)) represent a truevertical depths at the first and second measured depths.
 2. The methodaccording to claim 1, wherein: the tool string includes first, second,and third subsurface longitudinally spaced pressure sensors atcorresponding first, second, and third measured depths; (b) comprisescausing the first, second, and third pressure sensors to acquire first,second, and third annular drilling fluid pressure measurements; and (c)comprises processing the first, second, and third pressure measurementsto compute a first theoretical surface annular back pressure for awellbore interval between the first and second measured depths and asecond theoretical surface annular back pressure for a wellbore intervalbetween the second and third measured depths.
 3. The method according toclaim 1, wherein drilling fluid is being circulated in the tool stringand the wellbore is being drilled in (b).
 4. The method according toclaim 1, wherein drilling fluid is static in the tool string in (b). 5.The method according to claim 1, wherein the annular pressuremeasurements are acquired at a surface processor in (b) via a wireddrill pipe communications channel.
 6. The method according to claim 1,further comprising: (d) acquiring a surface annular back pressuremeasurement; and (e) adjusting a surface annular back pressure of thewellbore such that the measured surface annular backpressure in (d) issubstantially equal to the theoretical surface annular back pressurecomputed in (c).
 7. The method according to claim 6, wherein the surfaceannular back pressure is automatically adjusted in (e) via incrementallyadjusting a choke position.
 8. The method according to claim 6, furthercomprising: (f) repeating (b), (c), (d), and (e) in a control loop suchthat surface annular back pressure is automatically adjusted in (e) inresponse to changes in the theoretical surface annular back pressurecomputed in (c).
 9. The method according to claim 6, wherein: thesurface annular backpressure is automatically increased in response toan increase in the theoretical surface annular back pressure; and thesurface annular backpressure is automatically decreased in response to adecrease in the theoretical surface annular back pressure.
 10. Themethod according to claim 9, wherein the surface annular backpressure isautomatically increased in response to at least one of (i) a decrease incuttings density while drilling, (ii) borehole washout, (iii) formationfluid flowing into the wellbore, and (iv) drilling fluid changesresulting in a decreased interval density.
 11. The method according toclaim 9, wherein the surface annular backpressure is automaticallydecreased in response to at least one of (i) an increase in cuttingsdensity while drilling, (ii) a wellbore pack-off, (iii) lostcirculation, and (iv) drilling fluid changes resulting in an increasedinterval density.